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PROCEEDINGS 


OF   THE 


CAMBRIDGE  PHILOSOPHICAL  SOCIETY 

VOLUME   XIX 


CAMBRIDGE    :   PRINTED  BY 

J.  B.  PEACE,  M.A., 
AT  THE  UNIVERSITY  PRESS 


PROCEEDINGS 


OF  THE 


CAMBEIDGE  PHILOSOPHICAL 
SOCIETY 


VOLUME  XIX 

30  October  1916—24  November  1919 


CAMBRIDGE 

AT  THE  UNIVERSITY  PRESS 

AND  SOLD  BY 

DEIGHTON,  BELL  &  CO.  AND  BOWES  &  BOWES,  CAMBRIDGE 

CAMBRIDGE  UNIVERSITY  PRESS 
C.  F.  CLAY,  MANAGER,  FETTER  LANE,  LONDON,  E.C.  4 

1920 


CONTENTS. 

VOL.  XIX. 

PAGE 

A    self-recording   electrometer  for   Atmospheric   Electricity.     By   W.    A. 

Douglas  Rudge,  M.A.,  St  John's  College.   (Eleven  figs,  in  Text) .         1 

On  the  expression  of  a  number  in  the  form  ax^  +  by^  +  cz^  +  du'^.  By 
S.  Ramanujan,  B.A.,  Trinity  College.  (Communicated  by  Mr 
G.  H.  Hardy) H 

An  Axiom  in  Symbolic  Logic.  By  C.  E.  Van  Horn,  M.A.  (Communicated 

by  Mr  G.  H.  Hardy) 22 

A  Reduction  in  the  number  of  the  Primitive  Propositions  of  Logic.  By 
J.  G.  P.  NicoD,  Trinity  College.  (Communicated  by  Mr  G.  H. 
Hardy) 32 

Bessel  functions  of  equal  order  and  argument.   By  G.  N.  Watson,  M.A., 

Trinity  College 42 

The  limits  of  applicability  of  the  Principle  of  Stationary  Phase.    By 

G.  N.  Watson,  M.A.,  Trinity  College 49 

On  the  Functions  of  the  Mouth-Parts  of  the  Common  Pravni.    By  L.  A. 

Borradatlb,  M.A.,  Selwyn  College 56 

The  Direct  Solution  of  the  Quadratic  and  Cubic  Binomial  Congruences  with 

Prime  Moduli.   By  H.  C.  Pocklington,  M.A.,  St  John's  College  57 

On  a  theorem  of  Mr  G.  Polya.   By  G.  H.  Hardy,  M.A..  Trinity  College  .       60 

Submergence  and  glacial  climates  during  the  accumulation  of  the  Cambridge- 
shire Pleistocene  Deposits.  By  J.  E.  Marr,  Sc.D.,  E.R.S.,  St  John's 
College.   (Two  figs,  in  Text) 64 

071  the  Hydrodynamics  of  Relativity.    By  C.  E.  Weathers  urn,  M.A. 

(Camb.),  D.Sc.  (Sydney),  Ormond  College,  Parkville,  Melbourne      .       72 

0>i  the  convergence  of  certain  multiple  series.    By  G.  H.  Hardy,  M.A., 

Trinity  CoUege 86 

Bessel  functions  of  large  order.  By  G.  N.  Watson,  M.A.,  Trinity  College  .       96 

A  particular  case  of  a  theorem  of  Dirichlet.  By  H.  Todd,  B.A.,  Pembroke 
College.  (Communicated,  with  a  prefatory  note,  by  Mr  H.  T.  J. 
Norton) HI 


VI 


Contents 


On  Mr  Ramanujari' s  Empirical  Expansions  of  Modular  Functions.  By 
L.  J.  MoRDELL,  Birkbeck  College,  London.  (Communicated  by 
Mr  G.  H.  Hardy) .         .         . 


Proceedings  at  the  Meetings  held  dm-ing  the  Session  1916 — 1917    . 

Extensions  of  Abel's  Theorem  and  its  converses.  By  Dr  A.  Kienast,  Kiis 
nacht,  Ziirich,  Switzerland.    (Communicated  by  Mr  G.  H.  Hardy 

Sir  George  Stokes  and  the  concept  of  uniform  convergence.  By  G.  H.  Hardy, 
M.A.,  Trinity  College 

Shell-deposits  formed  by  the  flood  of  January,  1918.  By  Philip  Lake,  M.A. 
St  John's  CoUege 

75  the  Madreporarian  Skeleton  an  Extraptrotoplasmic  Secretion  of  th 
Polyps?  By  G.  Matthai,  M.A.,  Emmanuel  College,  Cambridge, 
(Communicated  by  Professor  Stanley  Gardiner)  . 

On  Reactions  to  Stimidi  iii  Corals.  By  G.  Matthai,  M.A.,  Emmanuel  Col- 
lege, Cambridge.  (Communicated  by  Professor  Stanley  Gardiner) 

Notes  on  certain  parasites,  food,  and  capture  by  birds  of  the  Common  Earwig 
(Forficula  auricularia).  By  H.  H.  Brindley,  M.A.,  St  John's  College 

Reciprocal  Relations  in  the  Them-y  of  Integral  Equations.  By  ]\Iajor  P.  A. 
MacMahon  and  H.  B.  C.  Darling 

Fish-freezing.   By  Professor  Stanley  Gardiner  and  Professor  Nuttall 

On  the  branching  of  the  Zygopteridean  Leaf,  and  its  relation  to  the  probable 
Pinna-nature  of  Gp'opteris  sinuosa,  Goeppert.    By  B.  Sahni,  M.A. 
Emmanuel  CoUege.   (Communicated  by  Professor  Seward) 

The  Structure  of  Tmesipteris  Vieillardi  Dang.  By  B.  Sahni,  M.A. 
Emmanuel  CoUege.    (Communicated  by  Professor  Seward)     , 

On  Acmopyle,  a  Monotypic  New  Caledonian  Podocarp.  By  B.  Sahni, 
M.A.,  Emmanuel  CoUege.   (Communicated  by  Professor  Seward) 

Proceedings  at  the  Meetings  held  during  the  Session  1917 — 1918    . 

On  Certai7i  Trigonometrical  Series  which  have  a  Necessary  and  Sufficient 
Condition  for  Uniform  Convergence.  By  A.  E.  Jolliffe.  (Com- 
municated by  Mr  G.  H.  Hardy) 

Some  Geometrical  Interpretations  of  the  Concomitants  of  Two  Quadrics. 
By  H.  W.  TxjRNBULL,  M.A.  (Communicated  by  Mr  G.  H.  Hardy)  . 

Some  properties  of  p  (n),  the  number  of  partitions  of  n.  By  S.  Ramanujan, 
B.A.,  Trinity  CoUege 

Proof  of  certain  identities  in  combinatory  analysis:  (1)  by  Professor  L.  J. 
Rogers;  (2)  by  S.  Ramanujan,  B.A.,  Trinity  CoUege.  (Communi- 
cated, with  a  prefatory  note,  by  JVIr  G.  H.  Hardy)  .... 


117 
125 

129 

148 

157 

160 
164 
167 

178 

185 

186 
186 

186 

187 

191 
196 
207 

211 


Contents  vii 


On  Mr  Ramanujan'' s  congruence  properties  of  p  [n).  By  H.  B.  C.  Darling 
(Communicated  by  Mr  G.  H.  Habdy) 

On  the  exponentiation  of  well-ordered  series.   By  Miss  Dorothy  Wrinch, 
(Communicated  by  Mr  G.  H.  Hardy)       .         .        .     -   . 

The  Gauss-Bonnet  Theorem  for  Multiply -Connected  Regions  of  a  Surface 
By  Eric  H.  Neville,  M.A.,  Trinity  College     .... 


217 
219 
234 


On  an  empirical  formula  connected  with  Goldbach's  Theorem.  By  N.  M. 
Shah,  Trinity  College,  and  B.  M.  Wilson,  Trinity  College.  (Com- 
municated by  Mr  G.  H.  Hakdy) 238 

Note  on  Messrs  Shah  and  Wilson's  paper  entitled:  'On  an  empirical  formula 
connected  ivith  Goldbach's  Theorem'.  By  G.  H.  Hardy,  M.A.,  Trinity 
College,  and  J.  E.  Littlewood,  M.A.,  Trinity  College     .         .         .     245 

The  distribution  of  Electric  Force  between  tivo  Electrodes,  one  of  which  is 
covered  ivith  Radioactive  Matter.  By  W.  J.  Harrison,  M.A.,  Fellow 
of  Clare  College.   (One  fig.  in  Text) 255 

The  conversion  of  saw-dust  into  sugar.   By  J.  E.  Purvis,  M.A.  .         .     259 

Bracken  as  a  source  of  potash.   By  J.  E.  Purvis,  M.A 261 

The  action  of  electrolytes  on  the  electrical  condtictivity  of  the  bacterial  cell 
and  their  effect  on  the  rate  of  migratioii  of  these  cells  in  an  electric  field. 
By  C.  Shearer,  Sc.D.,  F.R.S.,  Clare  College 263 

The  bionomics  of  Aphis  grossulariae  Kalt.,  and  Aphis  viburni  Schr.  By 
Maud  D.  Haviland,  Bathurst  Student  of  Newnham  College.  (Com- 
municated by  Mr  H.  H.  Brindley) 266 

Note  on  an  exp)eriment  dealing  with  mutation  in  bacteria.  By  L.  Don- 
caster,  Sc.D.,  King's  College.   (Abstract) 269 

Golourimeter  Design.  By  H.  Hartridge,  M.D.,  Fellow  of  King's  College, 

Cambridge.   (One  fig.  in  Text) 271 

The  Natural  History  of  the  Island  of  Rodrigues.  By  H.  J.  Snell  (Eastern 
Telegraph  Company)  and  W.  H.  T.  Tams.  (Communicated  by 
Professor  Stanley  Gardiner) 283 

Preliminary  Note  on  the  Life  History  of  Lygocerus  {Proctotrypidae), 
hyperparasite  of  Aphid  i  us.  By  Maud  D.  Haviland,  Fellow  of 
Newnham  College.   (Communicated  by  Mr  H.  H.  Brindley)  .        .     293 

Note  on  the  solitary  wasp,  Crabro  cephalotes.    By  Cecil  Warburton, 

M.A.,  Christ's  College 296 

Neon  Lamps  for  Stroboscopic  Work.  By  F.  W.  Aston,  M.A.,  Trinity 
College  (D.Sc.,  Birmingham),  Clerk-Maxwell  Student  of  the  Uni- 
versity of  Cambridge.   (One  fig.  in  Text)  .....     300 


viii  Contents 

PAGE 

The  pressure  in  a  viscous  liquid  moving  throiigh  a  channel  ivith  diverging 
boundaries.  By  W.  J.  Harrison,  M.A.,  Fellow  of  Clare  College, 
Cambridge.   (One  fig.  in  Text) 307 

The  Effect  of  Ions  on  Ciliary  Motion.    By  J.  Gray,  M.A.,  Fellow  of 

King's  College,  Cambridge        ........     313 

A  Note  on  Photosynthesis  and  Hydrogen  Ion  Concentration.    By  J.  T. 

Saunders,  M.A.,  Christ's  College 315 

The  distribution  of  intensity  along  the  loosiiive  ray  parabolas  of  atoms  and 
molecules  of  hydrogen  and  its  possible  explanation.  By  F.  W.  Aston, 
M.A.,  Trinity  College  (D.Sc.,  Birmingham),  Clerk-Maxwell  Student 
of  the  University  of  Cambridge.   (Three  figs,  in  Text)      .         .         .     317 

Gravitation  and  Light.    By  Sir  Joseph  Larmor,   St  John's  College, 

Lucasian  Professor 324 

On  a  Micro-voltameter.  By  C.  T.  R.  Wilson,  M.A.,  Sidney  Sussex  College     345 

The  self-oscillations  of  a  Thermionic  Valve.   By  R.  Whiddington,  M.A., 

St  John's  College 346 

Proceedings  at  the  Meetings  held  during  the  Session  1918 — 1919    .         .     347 

Index  to  the  Proceedings  with  references  to  the  Transactions         .         .     350 


PEOCEEDINGS 


OF   THE 


CAMBRIDGE    PHILOSOPHICAL 
SOCIETY 


VOL.   XIX.     PART   I. 

[Michaelmas  Term  1916.] 


ODamijrilJgc: 

AT    THE    UNIVERSITY    PRESS 
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DEIGHTON,  BELL  &  CO.,    LIMITED, 
AND  BOWES  &  BOWES,  CAMBEIDGE. 

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C.   F.   CLAY,    MANAGER,   FETTER  LANE,   LONDON,   E.G. 

1917 

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Bowes  or  Messrs  Deighton,  Bell  &  Co.,  Limited,  Cambridge. 

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PROCEEDINGS 

OF   THE 


A  self-recording  electrometer  for  Atmospheric  Electricity.  By 
W.  A.  Douglas  Rudge,  M.A.,  St  John's  College. 

[Received  18  October  1915.] 

In  the  course  of  the  writer's  work  on  the  local  variations  of  the 
atmospheric  potential  gradient,  the  need  was  felt  for  a  simple  self- 
recording  electrometer.  Most  of  those  in  use  are  costly  and  at  the 
same  time  rather  elaborate  in  construction.  A  new  arrangement 
has  therefore  been  devised  which  has  answered  the  purpose  in  view, 
and  as  the  apparatus  may  be  useful  in  other  directions  a  description 
is  now  given. 

It  has  been  shown*  that  very  considerable  variations  of  the 
normal  potential  gradient  are  produced  by  clouds  of  dust  raised  by 
the  wind,  etc. ;  and  also  by  clouds  of  steam  escaping  under  pressure 
from  steam  boilers f.  These  variations  are  very  sudden  and  do  not 
last  for  a  long  time,  so  that  an  instrument  used  for  recording  them 
must  be  fairly  quick  acting.  After  a  considerable  amount  of 
preliminary  work,  the  type  of  instrument  adopted  was  a  modified 
form  of  the  quadrant  electrometer,  the  record  being  photographed 
upon  a  piece  of  bromide  paper  attached  to  a  revolving  cylinder. 
One  special  use  to  which  the  electrometer  was  to  have  been  applied 
was  to  find  the  relation  between  the  potential  gradient  and  the 
altitude  of  the  place  of  observation,  and  for  this  purpose  it  was 
proposed  to  construct  ten  or  more  instruments,  so  that  a  number 
of  observations  could  have  been  carried  out  simultaneously.  Some 
work  of  this  kind  has  already  been  done  in  South  Africa  from 
which  it  appears  that  the  potential  gradient  near  to  the  ground 
diminishes  with  the  height  of  the  place  of  observation  above  sea 
level;]:.     In  order  to  get  satisfactory  results  it  is  necessary  for  the 

*  Proc.  Roy.  Sor.  A,  Vol.  90.  t  Proc.  Pflij.   Soc.  A,  Vol.  90. 

J   Trans.  Poy.   Soc.  SnntJi  Afrirn,  Vol.  vi,  pai-t  5. 

VOL.   XIX.   PT.   I.  1 


2  Mr  Ri(d[/e,  A  self- recording  electrometer 

stations  to  be  chosen  as  far  removed  as  possible  from  the  disturbing 
influence  of  manufacturing  operations,  and  of  railways,  and  it  was 
intended  to  have  taken  a  set  of  observations  in  the  neighbourhood 
of  the  Dead  Sea,  as  in  that  district,  stations  for  the  instruments 
might  have  been  chosen  with  altitudes  varying  from  1400  ft  below 
sea  level  to  3000  ft  above,  and  in  an  open  country.  As  a  number 
of  instruments  were  required  it  was  necessary  to  keep  the  cost  of 
construction  low,  and  this  has  been  achieved  in  the  instrument  to 
be  described,  so  that  the  cost  of  material  is  less  than  ten  shillings 
and  a  moderate  amount  only  of  mechanical  skill  is  required  in  the 
construction. 

The  complete  apparatus  consists  of: 

(1)  The  Electrometer. 

(2)  The  recording  cylinder. 

(3)  The  illuminating  arrangement. 

(4)  The  charging  battery. 

(5)  The  collecting  system. 

(1)  The  Electrometer.  This  consists  of  four  curved  pieces  of 
brass  cut  from  a  tube  of  3  cm.  diameter,  and  attached  to  a  block  of 
ebonite.  The  alternate  pieces  were  connected  together  in  the  usual 
way.  Each  conductor  subtended  an  angle  at  the  centre  of  the 
mirror  of  about  60°,  and  the  adjacent  conductors  were  about 
1  mm.  apart.  The  "  needle  "  was  formed  from  a  piece  of  silvered 
paper,  2*5  x  1"5  cm.  carrying  a  small  mirror,  or  a  piece  of  silvered 
thin  "cover"  glass  could  be  used  for  both  needle  and  mirror.  A  fine 
wire  was  attached  to  the  needle  to  support  a  piece  of  wire  gauze 
which  was  immersed  in  a  small  bottle  containing  paraffin  oil,  for 
damping  the  motion  of  the  needle.  The  system  was  suspended  by 
a  fine  phosphor  bronze  wire  by  means  of  which  the  needle  could  be 
charged,  Fig.  1.  The  whole  was  enclosed  in  a  thin  wooden  case 
having  a  small  window  in  front,  and  ebonite  plugs  to  allow  of 
connection  being  made  to  the  quadrants. 

(2)  The  recording  cylinder.  This  is  the  most  novel  feature  of 
the  instrument,  and  is  constructed  from  one  of  those  small  round 
clocks  which  may  be  bought  from  a  shilling  upwards.  Two  sizes 
of  clock-case  are  common,  of  diameters  10  cm.  and  6  cm.,  and  both 
of  these  sizes  have  been  used.  A  brass  tube  is  substituted  for  the 
hour  hand  at  the  fi'ont  of  the  clock,  and  a  similar  piece  of  brass 
tubing  is  attached  to  the  arbor  at  the  back  of  the  clock,  which  is 
attached  to  the  minute  hand  and  used  for  setting  the  clock.  These 
two  tubes  are  in  the  same  straight  line  and  furnish  a  convenient 
axis  about  which  the  clock  as  a  whole  can  rotate.  If  the  tube 
attached  to  the  hour  hand  is  fixed,  the  clock-case  will  turn  round 
once  in  twelve  hours,  whilst  if  the  minute  hand  is  fixed,  the  clock 


for  Atmospheric  Electriciti/ 


3 


rotates  once  in  one  hour.  Two  scales  of  measurement  are  thus 
possible  and  both  have  been  employed.  No  difficulty  was  found  in 
taking  twenty-four,  or  two  hour  records,  for  although  the  records 
overlapjjed  it  was  (|uite  easy  to  distinguish  one  part  from  the  other. 
A  light  zinc  tube  was  slipped  over  the  clock-case  to  give  a  good 
support  for  the  bromide  paper  which  was  wrapped  round  outside. 
The  whole  clock  was  made  to  balance  by  fastening  small  pieces  of 
lead  to  the  inside  of  the  case,  but  during  the  working  a  little 


Fig.  2. 
hour  hand  arbor  fixed  by  the  pin  P. 


Fig.  1. 
The  electrometer. 


irregularity  occurs  as  a  consequence  of  the  unwinding  of  the  spring; 
this  however  is  not  very  great  and  a  number  of  clocks  could  be 
made  to  keep  time  together.  The  recording  cylinder  was  enclosed 
in  a  light  tight  case  with  a  long  narrow  slit  in  front,  Fig.  2. 

(3)  Illuminating  system.  As  the  apparatus  was  used  out  of 
doors,  a  lamp  was  unsuitable  as  a  means  of  illumination,  so  that 
daylight  was  used  and  found  to  be  very  suitable.  The  electrometer 
and  recording  cylinder  were  placed  at  the  opposite  ends  of  a  light 
tight  box  measuring  20  x  17  x  14  cm.     A  hole  was  made  in  the 

1—2 


j\[r  Rudge,  A  self-recording  electrometer 


J L 


I        ,      v- 


\         I 
\       I 


u 


for  AtniospJterio  Electrlcitij  5 

top  of  one  of  the  ends  of  the  box,  and  covered  over  with  a  piece 
of  silvered  glass,  upon  which  a  fine  vertical  scratch — to  serve 
as  a  slit — had  been  made.  A  lens,  which  could  slide  upon 
a  rod  inside  the  box,  was  employed  to  project  the  light  upon 
the  electrometer  mirror,  whence  after  reflection  it  was  returned 
to  the  same  end  of  the  box  as  the  slit,  but  at  a  lower  level, 
and  fell  upon  the  horizontal  slit  in  the  case  of  the  recording 
cylinder.  By  this  means  a  point  of  light  impinged  upon  the 
bromide  paper,  and  as  the  latter  rotated,  traced  out  the  curve 
which  appears  after  developing  the  paper  in  the  usual  manner. 
Fig.  3. 

To  Collector 


Fig.  4.     E  electrometer.     R  recording  apparatus.     .S'  slit. 

(4)  Charging  battery.  In  using  the  electrometer  the  opposite 
pairs  of  quadrants  were  kept  charged  to  a  fixed  potential  by  means 
of  a  battery  of  the  small  Leclanche  cell  used  for  "  flash  "  lamps. 
These  cells  are  sold  in  sets  of  three  and  a  batch  of  eight,  giving 
about  35  volts,  is  quite  sufficient  for  atmospheric  observations.  The 
centre  of  the  battery  was  earthed.  The  complete  apparatus  is 
shown  in  Fig.  4. 

(5)  The  Collector.  This  consisted  of  a  small  plate  of  brass  coated 
with  a  radioactive  preparation.  The  plate  was  fixed  in  the  centre 
of  a  very  short  piece  of  brass  tubing  and  the  open  ends  of  the 


6  Mr  Rudge,  A  self-recordiiKj  electi'uineter,  etc. 

tubing  covered  over  with  wire  gauze,  so  as  to  prevent  loss  of  radium 
by  rubbing,  etc.;  Avhilst  allowing  it  to  take  up  the  potential  of  the 
air.  The  collecting  plate  was  supported  at  the  end  of  an  insulated 
wire,  and  at  such  a  height  above  the  ground  as  would  give  a 
deflection  suitable  to  the  sensibility  of  the  electrometer. 

Up  to  the  present  time  the  apparatus  has  been  used  for  the 
purpose  of  taking  records  of  the  variations  in  the  potential 
gradient,  due  to  the  presence  of  clouds  of  dust  raised  by  traffic  on 
the  roads,  or  to  the  variation  caused  by  the  steam  escaping  from 
passing  trains.     A  number  of  representative  curves  are  given. 

No.  1.  This  is  a  twelve  hour  record,  taken  at  a  station  on  the 
Gog  Magog  Hill  about  four  miles  fi'om  Cambridge,  and  so  far  from 
the  railway  and  roadway  that  traffic  had  no  disturbing  influence. 

No.  2  is  a  simultaneous  record  taken  in  Hills  Road  at  a  distance 
of  less  than  a  quarter  of  a  mile  from  the  railway,  so  that  every 
passing  train  shows  its  influence  in  increasing  the  positive  potential. 

Nos.  3  and  4  are  a  pair  of  simultaneous  hour  records,  three 
being  taken  at  Cherryhinton  reservoir,  and  four  at  about  300  yards 
from  the  railway.  The  "  peaks  "  in  the  latter  indicate  the  passing 
of  a  train. 

No.  5  is  a  one  hour  record  taken  on  Hills  Road  and  shows  the 
remarkable  influence  of  the  dust  raised  by  passing  vehicles.  Every 
vehicle,  even  an  ordinary  bicycle,  if  it  raises  dust,  disturbs  the 
normal  electrification.  Nos.  6  and  7  are  simultaneous  records 
taken  at  some  little  distance  from  the  road.  Nos.  8  and  9  were 
taken  near  the  "Long"  road  railway  crossing  and  show  the 
influence  of  passing  trains.  Nos.  10  and  11  are  a  pair  of  simul- 
taneous records,  10  being  taken  in  the  Railway  yard,  and  showing 
the  effect  of  passing  train  and  "shunting  "  operations ;  11  was  taken 
about  a  mile  away  from  the  line. 

All  the  potentials  indicated  are  positive  and  the  records  are 
reduced  in  reproduction,  but  an  equal  range  of  negative  potentials 
could  be  recorded,  as  only  one  half  of  the  width  of  the  photo- 
graphic paper  was  used  in  the  records  given.  The  sensibility  of 
the  instrument  may  be  changed  by  varying  the  number  of  cells  of 
the  charging  battery. 


220  volts      > 


220  voU^^ 


:z; 


^)vi^f^4^  r\/^^(^ 


/V*— *— " 


1   P.M.  2    P.M. 

^0.  5.     Variation  in  positive  potential  due  to  the  clouds  of  dust  raised  by  traffic  on  the  roads. 


1   P.M. 


No.  6.     Taken  simultaneously  with  5. 


2    P.M. 


1  P.M. 


No.   7.     Taken  simultaneously  with  5. 


V 

2    P.M. 


1.30   A.M.  10.30    A.M. 

No.  8.     Variation  in  potential  due  to  steam  from  passing  trains. 


2.15  P.M.  3.15  P.M. 

No.  9.     Variation  in  potential  due  to  steam  from  passing  trains. 


W^%#»jiw 


No.  10.     Variation  in  positive  potential  due  to  "shunting"  of  trains. 


No.  11.     Taken  simultaneously  with  No.  10,  but  at  a  distance  of  more  than  a  mile  from 

the  railway. 


Mr  Rainanujau,  On  t/ie  ej^pressiun  of  a  nv/inber,  etc.       11 

On  the  expression  of  a  number  in  the  form  ax-  +  by-  +  cz-  +  dv-. 
By  S.  Ramanujan,  B.A.,  Trinity  College.  (Communicated  by 
Mr  G.  H.  Hardy.) 

[Received  19  September  1916 ;  read  October  30,  1916.] 

1.  It  is  well  known  that  all  positive  integers  can  be  expressed 
as  the  sum  of  four  squares.  This  naturally  suggests  the  question  : 
For  what  positive  integral  values  of  a,  b,  c,  d  can  all  positive 
integers  be  expressed  in  the  form 

ax-  +  by-  +  cz-  +  du"  ?  (ll) 

I  prove  in  this  paper  that  there  are  only  55  sets  of  values  of 
a,  b,  c,  d  for  which  this  is  true. 

The  more  general  problem  of  finding  all  sets  of  values  of 
a,  b,  c,  d,  for  which  all  integers  luith  a  finite  number  of  exceptions 
can  be  expressed  in  the  form  (II),  is  much  more  difficult  and 
interesting.  I  have  considered  only  very  special  cases  of  this 
problem,  with  two  variables  instead  of  four ;  namely,  the  cases  in 
which  (I'l)  has  one  of  the  special  forms 

a{x^  +  y'-  +  z^)  +  bu-   (1-2), 

and  a{x'^ +y-)-\-b{z''-\-u?)    (1-3). 

These  two  cases  are  comparatively  easy  to  discuss.  In  this 
paper  I  give  the  discussion  of  (1"2)  only,  reserving  that  of  (1"3) 
for  another  paper. 

2.  Let  us  begin  with  the  first  problem.  We  can  suppose, 
without  loss  of  generality,  that 

a^b^c^d  (2'1). 

If  a  >  1,  then  1  cannot  be  expressed  in  the  form  (I'l) ;  and  so 

a  =  l  (2-2). 

If  b>  2,  then  2  is  an  exception ;  and  so 

1<6^2   (2-3). 

We  have  therefore  only  to  consider  the  two  cases  in  which  (1"1) 
has  one  or  other  of  the  forms  i 

X-  +  y-  +  cz^  +  du-,         X-  +  2y'-  +  cz-  +  dii-. 
In  the  first  case,  if  c>  3,  then  3  is  an  exception ;  and  so 

l^c^3  (2-31). 

In  the  second  case,  if  c  >  5,  then  5  is  an  exception ;  and  so 

2^c$5  (2-32). 

We  can  now  distinguish  7  possible  cases. 

(2-41)     x"- -\- y- +  z'- +  du?. 
If  rf  >  7,  7  is  an  exception ;  and  so 

X^d^l (2-411). 

(2-42)     X-  +  y-  +  2z-  +  du'. 


12 


Mr  Raniaiiujan,  On  tlie  expreasiuii  of  a  namber 


If  c?  >  14,  14  is  an  exception  ;  and  so 

2^cZ^14    (2-421). 

(2-43)    x"  +  y'  +  3^2  -1-  dii\ 
If  c?  >  6,  6  is  an  exception  ;  and  so 

'^^d^Q    (2-431). 

(2-44)     X'  +  ly""  +  ^z^  +  du-. 
li  d  >*1,  7  is  an  exception ;  and  so 

2^d^1 (2-441). 

(2-45)     x''  +  2?/-  +  '2>z'  +  diC'. 
If  (Z  >  10,  10  is  an  exception  ;  and  so 

3^c^^l0    (2-451). 

(2-46)    X-  -r  '±y''  +  ^z-  +  diJb^. 

If  c?  >  14;  14  is  an  exception ;  and  so 

4^c^^l4  (2-461). 

(2-47)     x'-^'ly-^-hz^^diC-. 

If  c?>  10,  10  is  an  exception ;  and  so 

o^cZ^lO  : (2-471). 

We  have  thus  eliminated  all  possible  sets  of  values  of  a,  b,  c,  d, 
except  the  following  55  : 


1, 

1 

2 

3, 

5 

1, 

2 

2 

4, 

5 

2, 

2 

2' 

5, 

5 

2, 

2 

li 

1, 

6 

1, 

3 

1, 

2 

6 

2, 

3 

2, 

2' 

6 

2, 

3 

1', 

3, 

6 

3, 

3 

2, 

3, 

6 

*3, 

3 

2 

4, 

6 

1, 

4 

2' 

5, 

6 

2, 

4 

1, 

1, 

7 

2, 

4 

1, 

2, 

7 

3, 

4 

2, 

2, 

7 

3, 

4 

2' 

3, 

7 

4, 

4 

2, 

4, 

7 

1, 

5 

2 

5, 

7 

2, 

5 

1^ 

2, 

S 

2, 

5 

2, 

3, 

8 

3, 

5 

,  2, 

4, 

9 

5, 

,  1, 

2, 

,  2, 

3, 

,  2, 

4, 

,  2, 

5, 

,  1, 

2, 

2, 

3, 

,  2, 

4, 

2 

5, 

,  1, 

9 

,  2, 

4, 

,  1, 

2, 

2 

4, 

,  1, 

2, 

,  2, 

4, 

,  1, 

2, 

,  2, 

4, 

8 

8 

9 

9 

9 

9 

10 

10 

10 

10 

11 

11 

12 

12 

13 

13 

14 

14 


in  the  for  in  ax-  +  by"  +  cz"  4-  da-  13 

Of  these  55  forms,  the  12  forms 

1,  1,  1,  2  1,  1,  2,  4  1,  2,  4,  8 

1,  1,  2,  2  1,  2,  2,  4  1,  1,  3,  8 

1,  2,  2,  2  1,  2,  4,  4  1,  2,  3,  6 

1,  1,  1,  4  1,  1,  2,  8  1,  2,  5,  10 

have  been  ah-eady  considered  by  Liouville  and  Pepin*. 

8.  I  shall  now  prove  that  all  integers  can  be  expressed  in  each 
of  the  55  forms.  In  order  to  prove  this  we  shall  consider  the  seven 
cases  (2*41) — (2*47)  of  the  previous  section  separately.  We  shall 
require  the  following  results  concerning  ternary  quadratic  arith- 
metical forms. 

The  necessary  and  sufficient  condition  that  a  number  cannot  be 
expressed  in  the  form 

x^-Vy--\-z'  (3-1) 

is  that  it  should  be  of  the  form 

4^(8ya  +  7),         (\  =  0,  1,2...,  /x  =  0,  1,2, ...) (3-11). 

Similarly  the  necessary  and  sufficient  conditions  that  a  numbei- 
cannot  be  expressed  in  the  forms 

x'+    tf+2z-    : (3-2), 

.T^+    f  +  '^z'    (3-3), 

.r 2  +  2 2/-  +  2^'^    ( 3  •  4 ) , 

a;-^  +  23/- +  3^2    (3-5), 

x- -^  "lif  +  ^z-    (3-6), 

^2+  2y^+  bz-    ...(3-7), 

are  that  it  should  be  of  the  forms 

4^(16/i  +  14)  (3-21), 

9M  9/*+    <))  (3-31), 

4^(  8/.+    7)  (3-41), 

4^(16ya  +  10)  (3-51), 

4^(16/^+14)  (3-61), 

25^(25/x+10)   or   25^(25/*  +  15)t  (3-71). 

"  There  are  a  large  number  of  short  notes  by  Liouville  in  vols,  v-viii  of  the 
second  series  of  his  journal.  See  also  Pepin,  ibid.,  ser.  4,  vol.  vi,  pp.  1-67.  The 
object  of  the  work  of  Liouville  and  Pepin  is  rather  different  from  mine,  viz.  to 
determine,  in  a  number  of  special  cases,  explicit  formulae  for  the  number  of 
representations,  in  terms  of   other  arithmetical  functions. 

t  Results  (3-11)— (3-71)  may  tempt  us  to  suppose  that  there  are  similar  simple 
results  for  the  form  ax-  +  hy-  +  cz-,  whatever  are  the  values  of  a,  b,  c.  It  appears, 
however,    that  in  most  cases  there   are  no  such  simple  results.      For  instance. 


14  Mr  Raiitavujan,  On  tJie  exjyression.  of  a  nninher 

The  result  concerning ./-  +  y^  +  z-  is  due  to  Cauchy  :  for  a  proof 
see  Landau,  Handhuch  der  LeJtre  von  der  Verteilung  der  Prim- 
zahlen,  p.  550.  The  other  results  can  be  proved  in  an  analogous 
manner.  The  form  x-  +  y"  +  ^z"-  has  been  considered  by  Lebesgue, 
and  the  form  x'^-\-y'^-{-'^Z'  by  Dirichlet.  For  references  see  Bach- 
mann,  Zahlentheorie,  vol.  iv,  p.  149. 

4.  We  proceed  to  consider  the  seven  cases  (2'41) — (2*47).  In 
the  first  case  we  have  to  show  that  any  number  N  can  be  expressed 
in  the  form 

N'  =  x-  +  y-  +  z-  +  du" (4- 1 ), 

d  being  any  integer  between  1  and  7  inclusive. 

If  JSf  is  not  of  the  form  4^(8yLt  +  7),  we  can  satisfy  (4-1)  with 
u  =  0.     We  may  therefore  suppose  that  iV^=  4^  (S/jl  +  7). 

First,  sup]3ose  that  d  has  one  of  the  values  1,  2,  4,  5,  6. 
Take  u  =  2\     Then  the  number 

N-du'  =  ^^(8fM+7-d) 

is  plainly  not  of  the  form  4^(8/4  +  7),  and  is  therefore  expressible 
in  .the  form  x^  +  y^  +  z^. 

Next,  let  d  =  S.     If  /i  =  0,  take  u  =  2\     Then 

N  -  dii-  =  4^+1. 

the  numbers  which  are  not  of  the  form  .r- +  2?/- + 10^-  are  those  belonging  to  one 
or  other  of  the  four  classes 

25^(8^  +  7),         25^(25^  +  5),         25^  (25/x  + 15) ,         25^  (25/^+20). 
Here  some  of  the  numbers  of  the  first  class  belong  also  to  one  of  the  next  three 
classes. 

Again,  the  even  numbers  which  are  not  of  the  form  x'^  +  ij-  +  lOz-  ai'e  the  numbers 

4^(16^  +  6), 
while  the  odd  numbers  that  are  not  of  that  form,  viz. 

3,  7,  21,  31,  33,  43,  67,  79,  87,   133,  217,  219,  223,  253,  307,  391,   ... 
do  not  seem  to  obey  any  simple  law. 

I  have  succeeded  in  finding  a  law  in  the  following  six  simple  cases: 
•«■''+   ?/2  +  4,--, 
X-+   y-  +  5z-, 
x^+   y'  +  6z', 
x^+   y^  +  8z-, 
x^  +  2y-^  +  Qz\ 
.r2  +  2y-  +  8^2. 
The  numbers  which  are  not  of  these  forms  are  the  numbers 
4^(8^  +  7)    or    (8^1  +  3), 
4^(8^  +  3), 
9^(9ya  +  3), 
4^(16/x+14),    (16m +  6),    or   (4^  +  3), 
4^  (8m +5), 
4^^(8^4-7)    or    (8^  +  5). 


in  the  form   (ur-  +  hy"  +  cz"  +  du?  15 

If  ya^l,  take«  =  2^+\     Then 

i\r-rf«;^  =  4^(8/^-5). 

In  neither   of  these   cases  is   TV  —  d}f-   of  the    form    4^  (8/i  +  7), 
and   therefore    in    either  case   it  can    be  expressed  in   the  form 

X-  +  2/2  4-  z". 

Finally,  let  d=l.  If  ^u,  is  equal  to  0,  1,  or  2,  take  «  =  2\ 
Then  N  -  d^ir  is  equal  to  0,  2 .  4^+\  or  4^+-.  If  /a^S,  take 
M.  =  2^+1.     Then 

iV-*/,^=4^(8/x-21). 

Therefore  in  either  case  N  —  du-  can  be  expressed  in  the  form 

a;2  +  2/2  4-  Z-. 

Thus  in  all  cases  N  is  expressible  in  the  form  (4'1).  Similarly 
we  can  dispose  of  the  remaining  cases,  with  the  help  of  the  results 
stated  in  §  3.  Thus  in  discussing  (2-42)  we  use  the  theorem  that 
every  number  not  of  the  form  (3'21)  can  be  expressed  in  the  form 
(3*2).  The  proofs  differ  only  in  detail,  and  it  is  not  worth  while 
to  state  them  at  length. 

5.  We  have  seen  that  all  integers  without  any  exception  can 
be  expressed  in  the  form 

m.  {x^  +'if  +  z^)  +  mi'^ (5-1), 

when  m  =  \,     li^n^l, 

and  m=  2,     n  =  1. 

We  shall  now  consider  the  values  of  m  and  n  for  which  all 
integers  with  a  finite  number  of  exceptions  can  be  expressed  in 
the  form  (5'1), 

In  the  first  place  7??  must  be  1  or  2.  For,  if  m  >  2,  we  can 
choose  an  integer  v  so  that 

7iu'  ^  V  (mod  7)1) 
for  all  values  of  u.     Then 

(nifx  +  v)  —  mi^ 
m 

where  fi  is  any  positive  integer,  is  not  an  integer ;  and  so  7nfj,  +  v 
can  certainly  not  be  expressed  in  the  form  (5'1). 

We  have  therefore  only  to  consider  the  two  cases  in  which  m  is 
1  or  2.     First  let  us  consider  the  form 

cc-  +  2/-  +  z-  +  nit^ (5'2). 

I  shall  show  that,  when  n  has  any  of  the  values 

1,  4,  9,  17,  25,  36,  68,  100 (5-21), 


16  Mr  lia/uanujan,  On  tJie  expression  of  a  nwniber 

or  is  of  any  of  the  forms 

4yt+2,     4yb  +  3,     8^•^-o,     16A;  +  12,     32/.^  + 20  ...(5-22), 
then  all  integers  save  a  finite  number,  and  in  fact  all  integers  from 
4?i  onwards  at  any  rate,  can  be  expressed  in  the  form  (5*2) ;  but 
that  for  the  remaining  values  of  n  there  is  an  infinity  of  integers 
which  cannot  be  expressed  in  the  form  required. 

In  proving  the  first  result  we  need  obviously  only  consider 
numbers  of  the  form  4*^  (8yu,  +  7)  greater  than  n,  since  otherwise 
we  may  take  w  =  0.  The  numbers  of  this  form  less  than  n  are 
plainly  among  the  exceptions. 

6.  I  shall  consider  the  various  cases  which  may  arise  in 
order  of  simplicity. 

(6-1)     7^  =  0  (mod  8). 
There  are  an  infinity  of  exceptions.     For  suppose  that 
N  =  Sfju  +  1. 
Then  the  number 

N  -  nu-  =  7  (mod  8) 
cannot  be  expressed  in  the  form  cc-  +  y-  +  z'^. 

(6-2)    n=2  (mod  4). 

There  is  only  a  finite  number  of  exceptions.  In  proving  this 
we  may  suppose  that  iV=4^(8/i  + 7).  Take  u=l.  Then  the 
number 

]SF  -  oiu^  =  4'"  (S/M  +  1)  -  n 
is  congruent  to  1,  2,  5,  or  6  to  modulus  8,  and  so  can  be  expressed 
in  the  form  x^  +  y^  +  z^. 

Hence  the  only  numbers  which  camiot  be  expressed  in  the 
form  (5"2)  in  this  case  are  the  numbers  of  the  form  4^(8/i+  7)  not 
exceeding  n. 

(6-3)     n=h  (mod  8). 

There  is  only  a  finite  number  of  exceptions.  We  may  suppose 
again  that  i\r  =  4^ (8/i  +  7).     First,  let  X  =|=  1 .     Take  u=\.     Then 

N  -  nu-  =  4^  (8/i  +  7)  -  n  =  2  or  3  (mod  8). 
If  X  =  1  we  cannot  take  u  =  l,  since 

N  -  n  =  7  (mod  8) ; 
so  we  take  u  =  2.     Then 

JSf-  nu-  =  V  (8/i  +  7)  -  4n  =  8  (mod  32). 

In  either  of  these  cases  N  —  nu^  is  of  the  form  cc'^  -\-y'^  +  z". 

Hence  the  only  numbers  which  cannot  be  expressed  in  the 
form  (5*2)  are  those  of  the  form  4^  (8/a  +  7)  not  exceeding  ??,  and 
those  of  the  form  4  (8/ti  +  7)  lying  between  n  and  4?l 


in  the  funii  aw-  +  Inf  +  cz-  +  da-  1'7 

(6-4)     /i=  3  (mod  4). 

There  is  only  a  finite  number  of  exceptions.     Take 
iV^=4M8/A  +  7). 
If  X^l,  take  ti=l.     Then 

N  —  nil-  =  1  or  5  (mod  8). 
If  X  =  0,  take  n  =  2.     Then 

N  -  nu?  =  3  (mod  8). 
In  either  case  the  proof  is  completed  as  before. 

In  order  to  determine  precisely  which  are  the  exceptional 
numbers,  we  must  consider  more  particularly  the  numbei'S  between 
n  and  4»  for  which  X  =  0.     For  these  \i  must  be  1,  and 

N  -nu-=  0  (mod  4). 
But  the  numbers  which  are  multiples  of  4  and  which  cannot  be 
expressed  in  the  form  .x-  +  y '  +  z'  are  the  numbers 

4''(8i.  +  7),         (/c  =  l,  2,  3,  ...,  v^O,l,  2,  3,  ...)• 
The  exceptions  required  are  therefore  those  of  the  numbers 

n  +  ¥{^v  +  1) (6-41) 

which  lie  between  n  and  4»  and  are  of  the  form 

8;i  +  7   (6-42). 

Now  in  order  that  (6'41)  may  be  of  the  form  (6'42),  k  must  be 
1  if  11  is  of  the  form  8A-  4-  3  and  k  may  have  any  of  the  values 
2,  3,  4,  ...  if  n  is  of  the  form  8A;+7.  Thus  the  only  numbers 
which  cannot  be  expressed  in  the  form  (5'2),  in  this  case,  are  those 
of  the  form  4^  (8/i  +  7)  less  than  n  and  those  of  the  form 

?i  +  4''(8i/+7),         (y-0,  1,  2,  3,  ...), 

lying  between  n  and  4?i,  where  k=\  if  n  is  of  the  form  8A;  +  3, 
and  K>\  if  ?r  is  of  the  form  8A;  +  7. 

(6-5)     n  =  1  (mod  8). 

In  this  case  we  have  to  prove  that 

(i)    if  n  ^  33,  there  is  an  infinity  of  integers  which  cannot  be 

expressed  in  the  form  (5"2) ; 
(ii)   if  n  is  1,  9,  17,  or  25,  there  is  only  a  finite  number  of 
exceptions. 
In  order  to  prove  (i)  suppose  that  iV  =  7  .  4^.     Then  obviously 
u  cannot  be  zero.     But  if  u  is  not  zero  n^  is  always  of  the  form 
4''(8t/+l).     Hence 

N  -  nu^  =  7  .  4^  -  ?i .  4"  (8v  +  1). 

Since  n  ^33,  X  must  be  greater  than  or  equal  to  k  +  2,  to  ensure 
that  the  right-hand  side  shall  not  be  negative.     Hence 

N  -  jui^  =  ^^  (Sk  +  7), 

VOL.  XIX.   PT.   I.  2 


18  31r  Ramanujan,  On  the  ex'pression  of  a   number 

where  k  =  14  .  V-''--  -  nv  -  ^  {n  +  7) 

is  an  integer ;  and  so  N  —  nu-  is  not  of  the  form  x-  +y'^  +  z\ 
In  order  to  prove  (ii)  we  may  suppose,  as  usual,  that 

N  =  4^  (Sfi  +  7). 
IfX  =  0,  take  w=l.     Then 

iV  -  nil''  =  8//  +  7  -  7?  =  6  (mod  8). 
If  X^l,  take  w=  2^-1.     Then 

where  k  =  4<  (fji  +  l) -^{n +  7). 

In  either  case  the  proof  may  be  completed  as  before.  Thus  the 
only  numbers  which  cannot  be  expressed  in  the  form  (5'2),  in 
this  case,  are  those  of  the  form  8/u.  +  7  not  exceeding  n.  In 
other  words,  there  is  no  exception  when  n  =  1  ;  7  is  the  only 
exception  when  n  =  9;  7  and  15  are  the  only  exceptions  when 
n  =  17  ;  7,  15  and  23  are  the  only  exceptions  when  n  =  25. 

(6-Q)     n  =  4  (mod  32). 
By  arguments  similar  to  those  used  in  (6'5),  Ave  can  show  that 
(i)    if  w  ^  132,  there  is  an  infinity  of  integers  which  cannot 

be  expressed  in  the  form  (5*2) ; 
(ii)    if  n  is  equal  to  4,  36,  68,  or  100,  there  is  only  a  finite 
number  of  exceptions,   namely  the  numbers  of  the 
form  4'^  (8yu,  +  7)  not  exceeding  n. 

(6-7)     ?i  =  20  (mod  32). 

By  arguments  similar  to  those  used  in  (6'3),  we  can  show  that 
the  only  numbers  which  cannot  be  expressed  in  the  form  (5'2)  are 
those  of  the  form  4^  (8yLi  +7)  not  exceeding  n,  and  those  of  the  form 
4^(8/A  +  7)  lying  between  n  and  4n. 

(6-8)     n=  12  (mod  16). 

By  arguments  similar  to  those  used  in  (6'4),  we  can  show  that 
the  only  numbers  which  cannot  be  expressed  in  the  form  (5"2)  are 
those  of  the  form  4^  (Sfi  +  7)  less  than  n,  and  those  of  the  form 

n  +  4>^(8v  +  1),         (i/  =  0,  1,  2,  3,  ...), 
lying  between  n  and  4>i,  where  /c  =  2  if  n  is  of  the  form  4  (8k  +  3) 
and  AC  >  2  if  w  is  of  the  form  4  (8A;  +7). 

We  have  thus  completed  the  discussion  of  the  form  (5  2),  and 
determined  the  exceptional  values  of  iV  precisely  whenever  they 
are  finite  in  number. 

7.     We  shall  proceed  to  consider  the  form 

2  (ic^  +y"  +  z')  +  mir .(7-1). 


in   the  fornt  (i.r-  +  bij'^  -\-  cz-  +  (hr  ID 

In  the  first  place  n  must  be  odd ;  otherwise  the  odd  iiuinbers 
cannot  be  expressed  in  this  form.  Suppose  then  that  n  is  odd. 
I  shall  show  that  all  integers  save  a  finite  number  can  be  expressed 
in  the  form  (7"1):  and  that  the  numbers  which  cannot  be  so 
expressed  are 

(i)    the  odd  numbers  less  than  n, 

(ii)    the  numbers  of  the  form  4^^  (16yu.  +  14)  less  than  4n, 
(iii)   the  numbers  of  the  form  n-\-  4^(16ya+  14)  greater  than 

n  and  less  than  9w, 
(iv)  the  numbers  of  the  form 

cn-\-¥{l(w+U\     (i^  =  0,  1,  2,  3,  ...), 

greater  than  9/i  and  less  than  25w,  where  c  =  1  if 
n  =  \  (mod  4),  c  =  9  if  w  =  3  (mod  4),  «  =  2  if  n^=  1 
(mod  16);  and  /c  >  2  if  /(-=9  (mod  16). 

First,  let  us  suppose  N  even.  Then,  since  n  is  odd  and  N  is 
even,  it  is  clear  that  u  must  be  even.     Suppose  then  that 

We  have  to  show  that  M  can  be  expressed  in  the  form 

x"->r  y-  -\-  Z'  +  27?,?'-  (7-2). 

Since  %i  =  2  (mod  4),  it  follows  from  (6'2)  that  all  integers  except 
those  which  are  less  than  2n  and  of  the  form  4-^  (8//.  +  7)  can  be 
expressed  in  the  form  (7*2).  Hence  the  only  even  integers  which 
cannot  be  expressed  in  the  form  (7'1)  are  those  of  the  form 
4^(16/*  + 14)  less  than  4n. 

This  completes  the  discussion  of  the  case  in  which  N  is  even. 
If  N  is  odd  the  discussion  is  more  difficult.  In  the  first  place, 
all  odd  numbers  less  than  n  are  plainl37^  among  the  exceptions. 
Secondly,  since  n  and  N  are  both  odd,  u  must  also  be  odd.  We 
can  therefore  suppose  that 

iV  =  ?i  +  2il/,     xv"  =  1  +  8A, 

where  A  is  an  integer  of  the  form  |^•(^•  +  l),  so  that  A  may 
assume  the  values  0,  1,  3,  6,  ....  And  we  have  to  consider 
whether  n  +  2if  can  be  expressed  in  the  form 

2  0r2  +  2/'^  +  ^'^)  +  w(l  +8A), 
or  M  in  the  form 

«-  +  2/-  +  ^-  +  4nA (7-3). 

If  M  is  not  of  the  form  4^  (8/a  +  7),  we  can  take  A  =  0.  If  it  is 
of  this  form,  and  less  than  4?i,  it  is  plainly  an  exception.  These 
numbers  give  rise  to  the  exceptions  specified  in  (iii)  of  section  7. 
We  may  therefore  suppose  that  M  is  of  the  form  4^  (8^*  +  7)  and 
greater  than  4/?. 

2—2 


20  Mr  Ramamtjan,  On  the  eo;pression  of  a   number 

8.  In  order  to  complete  the  discussion,  we  must  consider 
the  three  cases  in  which  n  =  1  (mod  8),  n  =  5  (mod  8),  and 
n  =  S  (mod  4)  separately. 

(8-1)     /I-  1  (mod  8). 
If  X  is  equal  to  0,  1,  or  2,  take  A  =  1.     Then 
M  -  4?iA  =  4^  (8/i  +  7)  -  4?i 
is  of  one  of  the  forms 

8z;  +  3,     4  (Sv  +  3),     4  {8v  +  6). 

If  A.  ^  3  we  cannot  take  A  =  1,  since  if  —  4?iA  assumes  the 
form  4  (8i/  +  7) ;  so  we  take  A  =  3.     Then 

M -  4n A  =  4^  (SyLt  +  7)  -  12n 

is  of  the  form  4  {8v  +  5).  In  either  of  these  cases  M  —  4nA  is  of 
the  form  x^  +  y'^  -\-  z^.  Hence  the  only  values  of  M,  other  than 
those  already  specified,  which  cannot  be  expressed  in  the  form 
(7*3).  are  those  of  the  form 

4«(8i;  +  7),        (z/  =  0,  1,2,  ...,«>2), 

lying  between  4?i  and  12??.  In  other  words,  the  only  numbers 
greater  than  9'n  which  cannot  be  expressed  in  the  form  (71),  in 
this  case,  are  the  numbers  of  the  form 

n+4«(8j;  +  7),   (i^  =  0,  1,  2,  ..., /c>  2), 

lying  between  9?i  and  25?l 

(8-2)     ?i  =  5  (mod  8). 
If  X  4=  2,  take  A  =1.     Then 

ili  -  4wA  =  4^  (8/i  +  7)  -  4?i 
is  of  one  of  the  forms 

8i/  +  3,      4  (8z/  +  2),     4  (8z/  +  3). 

If  \  =  2,  we  cannot  take  A  =  l,  since  ilf— 4?iA  assumes  the 
form  4  (8v  +  7) ;  so  we  take  A  =  3.     Then 

M-  4mA  =  4^  (8/A  +  7)  -  \%i 

is  of  the  form  4  (8y  +  5).  In  either  of  these  cases  M  —  4/?  A  is  of 
the  form  (x?  ■\-y'^  A-  z^.  Hence  the  only  values  of  M,  other  than 
those  already  specified,  which  cannot  be  expressed  in  the  form 
(7-3),  are  those  of  the  form  16  (8^t  +  7)  lying  between  4n  and  12?i. 
In  other  words,  the  only  numbers  greater  than  2n  which  cannot 
be  expressed  in  the  form  (7"1),  in  this  case,  are  the  numbers  of  the 
form  n  +  4"^(16/ti  +  14)  lying  between  9??  and  2.5». 


in  the  form  ax~  +  by-  +  oz-  -f  dti^  21 

(8-3)     n  =  3  (mud  4). 
If  \  =1=1,  take  A  =  1.     Then 

M  -  4»,A  -  4^  (8ya  +  7)  -  hi 
is  of  one  of  the  forms 

Si. +  3,    4(4i/+l). 
If  \  =  1,  take  A  =  3.     Then 

M  -  4mA  =  4  (8/*  +  7)  -  12/i 

is  of  the  form  4(4^-  +  2).  In  either  of  these  eases  M  —  4?iA  is  of 
the  form  j/-^  +  -tf  +  z-. 

This  completes  the  proof  that  there  is  only  a  finite  number  of 
exceptions.  In  order  to  determine  what  they  are  in  this  case,  we 
have  to  consider  the  values  of  M,  between  4?i  and  12w,  for  which 
A  = 1  and 

M  -  4h  A  =  4  (8;i  +  7  -  n)  =  0  (mod  16). 

But  the  numbers  which  are  multiples  of  16  and  which  cannot  be 
expressed  in  the  form  x-  +  y^  -\-  z-  are  the  numbers 

4''(8z/+7),       (/c  =  2,  8,  4,  ...,  y=0,  1,  2,  ...)• 

The  exceptional  values  of  M  required  are  therefore  those  of 
the  numbers 

47«  +  4*^  (8i.  +  7) (8-31) 

which  lie  between  4fi  and  \2n  and  are  of  the  form 

4(8/iA  +  7) (8-32). 

But  in  order  that  (831)  may  be  of  the  form  (8"32),  k  must  be 
2  if  n  is  of  the  form  8/^'  +  3,  and  k  may  have  any  of  the  values 
3,  4,  5,  ...  if  n  is  of  the  form  8A;  +  7.  It  follows  that  the  only 
numbers  greater  than  9«  which  cannot  be  expressed  in  the  form 
(7"1),  in  this  case,  are  the  numbers  of  the  form 

9w  +  4«  (16i^  +  14),  {v  =  0,  1,  2,  . . .), 

lying  between  9w  and  25?i,  where  k=2  if  n  is  of  the  form  8/v  +  3, 
and  /c  >  2  if  ?i  is  of  the  form  8^-  +  7. 

This  completes  the  proof  of  the  results  stated  in  section  7. 


22  Mr   Van  Horn,  An  Axiom  in  Symbolic  Logic. 


An  Axiom  in  Symbolic  Logic.     By  C.  E.  Van  Horn,  M.A. 
(Commimicated  by  Mr  Q.  H.  Hardy.) 

[Received  30  August  1916:   read  30  October  1916.] 

Philosophy's  task  is  a  search  for  the  primal  and  fundamental 
elements  of  the  world.  Its  face  is  turned  in  the  opposite  direction 
to  that  of  science  and  mathematics.  Philosophy  hands  back  to 
them  its  results,  and  they  as  best  they  can  construct  systematic 
bodies  of  doctrine  that  purport  to  show  us  what  the  world  may  bo 
on  the  one  hand  (science)  and  what  the  world  might  be  on  the 
other  (mathematics).  As  philosophy  advances  in  the  pursuit  of  its 
task  it  is  continually  vacating  old  ground  to  science  and  mathe- 
matics. The  history  of  this  change  of  boundary  can  be  traced  in 
the  changes  in  the  nomenclature  of  human  knowledge :  Natural 
Philosophy  has  become  Physics ;  Mental  Philosophy  has  become 
Psychology ;  Moral  Philosophy  is  becoming  the  inductive  science 
of  Ethics.  Thus  (paradoxically  speaking)  philosophy's  advance  is 
to  be  marked  by  the  retreat  of  her  boundaries. 

It  is  interesting  to  Avatch  this  retreat  in  a  field  occupied  b}' 
philosophy  from  its  very  beginning,  and  until  recently  supposed  to 
be  its  permanent  possession.  I  refer  to  the  field  of  the  foundations 
of  mathematics.  Here  large  areas  once  occupied  by  philosophy  by 
sovereign  right  of  long  control  are  slowly  passing  into  the  possession 
of  pure  mathematics;  and  by  the  way  both  are  gainers  by  the 
transfer*. 

To  facilitate  the  mathematical  treatment  of  these  new  areas  a 
new  instrument  of  investigation  had  to  be  invented,  namely,  Mathe- 
matical, or  Symbolic,  Logic.  This  new  logic,  which  is  infinitely 
more  powerful  than  the  traditional  logic,  and  which  embraces  all 
that  is  really  self-consistent  in  the  old  logic,  makes  possible  a 
precise  and  easy  handling  of  all  the  highly  abstract  and  complex 
ideas  occurring  in  the  noAv  fields.  For  example,  both  philosophy 
and  the  old  logic  found  themselves  involved  in  many  a  tangle  on 
questions  concerning  classes  and  relations  because  neither  possessed 
the  requisite  instruments  of  analysis.  Again,  philosophy  had 
wandered  into  a  veritable  labyrinth  of  difficulties  concerning 
infinity,  quantity,  continuity,  and  so  on.  Here  too  the  secret  of 
the  trouble  lay  in  the  inadequacy  of  the  instruments  of  analysis 
afforded  by  the  traditional  logic. 

*  Much  valuable  light  is  thrown  upon  the  details  of  this  process  in  the  writings 
of  Bertrand  Russell,  especially  in  the  preface  and  introductory  chapters  of  the 
Frincipia  Mathematica,  Vol.  i.  1910;  and  more  recently  in  his  Scientific  Method  in 
Philosophy,  1914. 


Ml'   Van  Horn,  An  Axiom  in  Symbolic  Logic  28 

Nuw  however  the  matter  is  all  changed.  Philosophy,  equipped 
with  the  latest  instruments  of  mathematical  logic,  is  able  to  deal 
successfidly  with  the  problems  of  these  fields.  In  fact  so  fully  have 
these  ideas  been  analysed  that  at  last  philosophy  as  such  has 
relinquished  these  fields  to  pure  mathematics.  Even  more,  the 
whole  field  of  deduction  has  now  become  the  foundation-branch  of 
mathematics  and  has  developed  into  a  precise  Calculus  of  Pro- 
positions. Out  of  it  grow  by  easy  stages  the  Calculus  of  Classes 
and  the  Calculus  of  Relations,  and  these  in  turn  grow  by  equally 
easy  stages  into  all  the  manifold  branches  of  pure  mathematics  as 
more  commonly  known.  It  is  in  these  and  similar  ways  that 
philosophy  and  pure  mathematics  are  both  gainers  by  the  transfer 
of  the  fields  recently  acquired  by  mathematics  from  philosophy. 

It  is  now  easy  to  understand  why  the  axioms  of  mathematical 
logic  (and  so  of  all  pure  mathematics)  lie  in  the  borderland  between 
philosophy  and  mathematics,  and  are  thus  the  concern  of  the 
philosopher  equally  with  the  mathematician.  To  depart  entirely 
from  our  figures  and  adopt  others,  the  rootage  of  mathematics  is  in 
philosophy.  It  is  here  too  that  we  meet  the  innovations  of  mathe- 
matical logic  that  appear  so  fantastic  to  the  philosopher  trained 
only  in  the  old  logic.  Its  definitions  and  treatment  of  some  of  the 
common  terms  of  language  seem  so  at  variance  with  what  the 
traditional  logician  is  familiar  with  that  he  often  views  the  new 
logic  as  the  victim  of  some  delusion.  It  appears  however  from  the 
nature  of  the  case  itself  that  many  of  those  peculiarities,  which 
from  the  view-point  of  traditional  logic  would  be  described  as 
abnormal,  do  not  deserve  to  be  so  described ;  that  in  fact  it  is  in 
the  theories  of  the  traditional  logician  and  philosopher  that  the 
abnormalities  really  occur*. 

In  order  to  indicate  what  seems  to  me  a  possible  simplification 
of  the  axiomatic  basis  of  mathematical  logic  I  wish  to  introduce  in 
a  new  form  an  idea  advocated  by  Shelfer.  Its  importance  lies  in 
the  fact  that  in  terms  of  it  Sheffer  was  able  to  define  the  four 
fundamental  operations  of  logic,  namely.  Negation,  Disjunction, 
Implication,  and  Conjunction  or  Joint  Assertion.  It  is  a  familiar 
fact  that  Kronecker  found  the  use  of  certain  auxiliary  quantities 
(let  us  call  them  '  parameters ')  of  great  value  in  his  algebraic 
investigations,  the  chief  value  lying  in  the  fact  that  their  dis- 
appearance led  to  desired  relations  among  numbers  essential  to  his 
investigations.  It  is  a  precisely  similar  use  of  Sheffer's  idea  that 
I  desire  to  make  in  the  field  of  the  philosophy  of  logic.  In  terms 
of  it  I  define,  after  him,  the  four  fundamental  operations  of  logic. 
Then,  unlike  him,  I  work  by  means  of  an  axiom  to  eliminate  that 
idea  from  the  formulae,  and  in  so  doing  to  arrive  at  the  desired 

*  Cf.  Russell,  Scientific  Method  in  Philosophij,  chap.  i. 


24  Mr   Van  Horn,  An  Axiom  in  Si/ntholic  Logic 

properties  and  relations  of  the  four  fundamental  operations.  The 
chief  excellence  of  my  method  seems  to  reside  in  the  fact  that 
proceeding  as  indicated  above  I  have  been  able  to  prove  as  pro- 
positions of  mathematical  logic  some  of  the  axioms  hitherto  laid 
down  at  the  basis  of  this  logic. 

In  its  most  satisfactory  form  the  axiomatic  basis  of  mathe- 
matical logic  has  been  stated  by  Bertrand  Russell  in  the  first 
volume  of  the  Principia  Mathematical.  In  *1  of  Vol.  i.,  pp.  98-101 , 
of  the  Principia  will  be  found  the  primitive  propositions  required 
for  the  theory  of  deduction  as  applied  to  elementary  propositions. 
I  confine  myself  to  these  purposely,  for  it  is  here  that  I  have 
succeeded,  I  believe,  in  simplifying  the  axiomatic  basis  of 
mathematical  logic. 

Let  p  and  q  be  any  two  elementary  propositions.  The  four 
fundamental  operations  give  us  (1)  ~  p  {not-p),  (2)  pv  q  (either  p 
or  q),  (3)  j9  D  q  (p  implies  q),  and  {4<)  p  •  q  (both  p  and  q).  After 
Sheffer,  I  define  these  four  results  in  terms  of  a  single  undefinable 
operation.  I  will  call  this  undefinable  operation  Deltation.  The 
result  of  performing  this  operation  upon  two  elementary  propositions 
p  and  q  is  symbolized,  after  Sheffer,  'pAq'  (read  " j)  deltas  q'). 
The  four  fundamental  operations  of  logic  can  be  expressed  as 
logical  functions  of  this  parameter  thus : 

Negation:  ~p.  =  .jjAj9  D£ 

Disjunction :  /;vg.  =  .~^jA~g  Df 

Implication  :  pD  q  .  =  .p  A  <^  q  Di 

Conjunction  :  p  •  q  .  =  .  '^  (p  A  q)  Df. 

These  definitions  of  the  four  fundamental  operations  of  logic 
as  functions  of  the  one  undefined  parameter,  Deltation,  are  made 
relevant  to  our  discussion  by  means  of  the  following  axiom. 

Axiom.  If  p  and  q  are  of  the  same  truth-value,  then  ' p  A  q  ' 
is  of  the  opposite  truth-value ;  but  if  j)  and  q  are  of  ojjposite  truth - 
values,  then  ' p  A  q'  is  true. 

For  convenience  of  reference  it  might  be  well  for  me  to  state  at 
this  point  Russell's  primitive  propositions  concerning  elementary 
propositions  as  he  enunciates  them  in  *1  of  the  first  volume  of 
the  Principia. 

*1.1  Anything  implied  by  a  true  elementary  proposition 
is  true.     Pp|. 

t  Whitehead  and  Russell,  Princiina  Mathematica,  Vol.  i.  1910,  Vol.  ii.  191'2, 
Vol.  III.  1913  (Cambridge  University  Press). 

X  Eussell  uses  the  letters  "Pp"  to  stand  for  "  primitive  proposition, "  as  does 
Peano. 


3fr   Van  Horn,  An  Axiom  in  Sipnbolic  Lo(jio  25 

*1.H  When  <^x  can  be  asserted,  where  x  is  a  real  variable, 
and  '  (fjxD  yfr  x '  can  be  asserted,  where  x  is  a  real  variable/then  yjrx 
can  be  asserted,  where  x  is  a  real  variable.     Pp. 

*1.2  h :  pvp.D  .p         Pp. 

*1.3  \- :  q.D  .pv  q         Pp. 

*  1.4  [■ :  pv  q  .0  .qv  p  Pp. 

*  1 .5  \- :  py/  (qv  r).D  .qv  {pv  r)    Pp. 

*1.6  h:  .q'^r.D-.pvq.D.pyr   Pp. 

*1.7  If  j9  is  an  elementary  proposition,  ~  p  is  an  elementary 
proposition.     Pp. 

*1.71  If  p  and  (/  are  elementary  propositions  'pvq'  is  an 
elementary  proposition.     Pp. 

*1.72  If  ^p  and  i/rj?  are  elementary  prepositional  functions 
which  take  elementary  propositions  as  arguments, '  (f)  pv  -ylrp'  is  an 
elementary  prepositional  function.     Pp. 

These  are  all  the  primitive  propositions  that  are  needed  for  the 
development  of  the  theory  of  deducti(jn,  as  applied  to  elementary 
propositions,  according  to  Russell's  method  of  treatment. 

It  is  my  purpose  to  show  that  by  means  of  my  axiom 
Russell's  primitive  propositions  *1.2  to  *1.7l  can  be  demon- 
strated. I  do  this  by  starting  at  the  very  beginning  and 
developing  the  immediate  consequences  of  three  of  the  axioms 
which  I  lay  down  as  the  basis  of  the  theory  of  deduction  as  applied 
to  elementary  propositions.  The  resulting  deductive  development 
at  length  reaches  a  point  where  it  includes  among  its  theorems 
Mr  Russell's  seven  pi'imitive  propositions  and  two  others  that  can 
take  the  place  of  his  definitions  of  Implication  and  Conjunction. 
Altogether  I  prove  seventeen  theorems.  Some  of  these  theorems 
occur  as  propositions  in  the  first  volume  of  the  Principia.  Al- 
though many  more  theorems  can  be  proved  as  simply  as  the  ones 
given,  to  economize  space  I  shall  stop  at  the  point  where  my 
development  of  Mathematical  Logic  includes  the  nine  theorems 
mentioned  above. 

I  will  now  state  the  three  axioms  used  in  this  paper.  The 
first  is  *  1.1  given  above,  the  last  is  my  axiom  as  already  enunciated. 

Axiom  1.  Anything  implied  by  a  true  elementary  proposition 
is  true. 

Axiom  2.  Ifp  and  q  are  elernentary propositions, then  " p  Aq' 
is  an  elementary  proposition. 

Axiom  3.  If  p  cund  q  are  of  the  same  truth-value,  then  '  p  Aq' 
is  of  the  opposite  truth -value ;  hut  if  p  and,  q  are  of  opposite  truth- 
values,  then  ' p  Aq'  is  true. 


26  Mr   Van  Hum,  An  Aadoiii  in  Symholic  Logic 

Theorem  1 

If  })  is  an  elementary  proposition,  ~  p  is  an  elementary  pro- 
position, 

Deni. 

Axiom  2  gives  us  ' p  Ap'  elementaiy  when  j)  is  elementary ; 
'pAp'  is  ~  p,  by  Definition  of  Negation.     Hence  the  theorem. 

This  is  a  proof  of  Mr  Russell's  primitive  proposition  *1.7  given 
above. 

Theorem  2 

Ifj}  and  q  are  elementary  ])ropositions,  ' pv (j'  is  an  elementary 
proposition. 

Dem. 

By  Theorem  1 ,  if  p  and  q  are  elementary  so  also  are  ~  p  and 
~  q.  Therefore,  by  Axiom  2, '  -^  p  A  ~  (/ '  is  elementary  ;  but  this, 
by  Definition  of  Disjunction,  is  ' pv q'.     Hence  the  theorem. 

This  is  Mr  Russell's  primitive  proposition  *1.7l  quoted  above. 

Theorem  3 
The  propositions  p  and  ~  p  are  of  opposite  truth-values. 

Dem. 

Two  possibilities  can  occur : 

1°:^  true.  By  Axiom  S,  " p  A  p'  is  false;  but  this  by 
Definition  of  Negation  is  ^  p;  hence  in  this  case  p  and  ~  p  are 
opposite  in  truth-value. 

2° :  j9  false.  By  Axiom  3,  'pAp'  is  true;  but  this  by 
Definition  of  Negation  is  ~  jo ;  hence  in  this  case  also  ^j  and  f^  p 
are  opposite  in  truth-value.     Hence  the  theorem. 

This  theorem  states  in  precise  form  the  information  usually 
given  in  text-books  on  logic  in  more  or  less  vague  statements  that 
are  called  '  definitions '  of  negation. 

Theorem  4 
\-.  pDp. 

Dem. 

[Th.  3]  h.  p  and  <^  p  of  opposite 

truth-values     (1) 

[(1).     Ax.  3]  \-.  pA  r^  p  (2) 

[(2).     Def.  of  Implication]  h.  theorem. 

This  is  proposition  *2.08f  of  the  Principia. 

I    0]}.  cit.  Vol.  I.  p.  105. 


Mr    Van  Horn,  An  Axiom  in  Symholio  Loyic  27 

Theorem  5 

If  2)  is  false,  ' p  A  q'  is  always  true. 

Bern. 

Two  possibilities  can  occur :  either  q  true,  or  q  false.  In  either 
case  ' p  A  q'  IB  true  by  Ax.  3. 

Theorem  6 

If  q  Is  false,  '  2J  A  q'  is  alivays  true. 
Proof  similar  to  that  of  preceding  theorem. 

Theorem  7 
llie  jyropositions  '  p  A  q'  and  '  q  A  p'  Jmve  the  same  triUh-ualue. 

Deni. 

li' p  and  q  are  of  the  same  truth- value  then,  by  Ax.  o,  ' p  A  q' 
and  '  q  A  p'  are  both  of  the  opposite  truth-value.  If  p  and  q  are 
of  opposite  truth-values  then,  by  Ax.  3,  '  p  A  q'  and  '  q  A  p'  are 
both  true.     Hence  the  theorem. 

Theorem  8 
The  proposition 

f"^  p  A  1^  {(^  q  A  f^  r) 

is  true  if  any  one  or  more  of  the  propositions  p,  q,  r  are  true;  but 
if  all  of  these  propositions  are  false  then  the  proposition 

~  jj  A  '^  {^  q  A  ~  r) 
is  false. 

Dem. 

Eight  possibilities  can  occur : 

1° :  p,  q,  r  all  true.  Then  (Th.  3)  ~  ^j,  ~  q,  ~  r  are  all  false. 
Hence  (Ax.  3) '  ~  9  A  ~  r '  is  true.  Hence  (Th.  3)  ~  (~  g-  A  ~  r) 
is  false.  Hence  (Ax.  3)  the  proposition  '~  jo  A  ~  (~  </  A  ~  /•)' 
is  true  in  this  case. 

2^ :  jj  and  q  true,  but  r  false.  By  Th.  S,  r^  p  and  ~  q  are 
false,  while  ~  r  is  true.  Hence  (Ax.  3)  '  ~  r/  A  ~  ?■ '  is  true. 
Hence  (Th.  3)  ~  (~  (/ A  ~  r)  is  false.  Hence  (Ax.  3)  the 
proposition  is  true  in  this  case.  In  a  similar  manner  in  the 
following  cases : 

3° :  j)  true,  q  false,  r  true ; 

4° :  ])  false,  q,  r  true ; 

o" :  p  true,  q,  r  false ; 

6° :  J)  false,  q  true,  r  false  ; 

7° :  p,  q  false,  r  true  ; 
we  have  '  ^'  j)  A  ^  (■--  </  A  ^^  /•) '  true. 

But  in  8" :  p,  q,  r  false,  we  have  ~  jj,  r^  q,  ^  r  all  true,  by 


28  Jllr   Van  Horn,  An  Axiont  in  H[iinholic  Logic 

Th.  8.    Hence  (Ax.  3) '~  </  A  '^  ?• "  is  false,  making  ~  (~  (/  A  ~  /•) 
true  (Th.  3).     Hence  (Ax.  3)  in  this  case  the  proposition  is  false. 

Hence  the  theorem. 

Theorem  9 
The  propositions 

'  <>•'  p  A  f^  (<^  q  A  f^  r)',    'r^(/A~(~^:>A~  r) ', 

always  have  the  same  truth-valm. 

This  follows  at  once  from  Th.  8. 

At  this  point  I  introduce  Mr  Russell's  definition  of  Equivalence  f 
as  it  occurs  in  the  Principia. 

Equivalence:  p  =  q.  =  .pDq.qDp         Df. 

Theorem  10 

h.  p=  <^  (^  p). 

Dem. 

We  first  prove  h.  jj  D  ~  ( ~  p).     Two  cases  arise  : 
1°:  p  true.     By  Theorem  3,  '^  ^  is  false,  ~  (~p)  is  true,  and 
f^  [f^  {^  py]  is  false.     Hence 

[Ax.  3]  h.  I?  A  ~  [~  (~  jj)]  (1) 

[(1).     Def.  Implica.]  Kj9D~(~p)  (2) 

2° :   p  false.     By  Th.  3,   r^  p  is  true,  ^^  {<^  p)  is  false,  and 
,^  [r^  ('^i^)]  is  true. 


[Ax.  3] 

h. 

j5  A  ~  [~  {"^  p)\  (3) 

[(1).     Implica.] 

V. 

p  D  (^  (<^  p)           (4) 

Hence  in  all  cases  we  have 

V. 

p'^r^{<^p)          (5) 

We  now  prove 

V. 

~(~_p)Di9. 

[Th.  3] 

V. 

q  and  ~  g  of  opposite 
truth-values         (6) 

[(6).     Ax.  3] 

h. 

r^qAq                  (7) 

[(7).      /] 

[(8).  Def.  Implica.] 

V. 

'^   {r^  p)   A    r^  p         (8) 

V. 

~(~p)Dp           (9) 

[(5).  (9).  Def.  Equiv.] 

V. 

theorem. 

This  is  proposition  *"4.13  of  the  Principia^.  It  is  the  Principle 
of  Double  Negation,  and  asserts  that  any  proposition  is  logically 
equivalent  to  the  denial  of  its  negation. 

t  Op.  cit.  Vol.  I.  p.  120,  *4.0l. 
%  Op.  cit.  Vol.  I.  p.  122. 


^fr   Van  Horti,  An  A.iiom,  in  Si/)iibolic  Logic  29 

Theorem  11 
H:  pvp .  D  .p. 

Dem. 

[Ax.  3]        ■  h.  ~  jj    and    '  <^  p  A  f^  p'  of 

opposite  truth- values      (1) 

[(1).  Ax.  3]  |-:~_p  A  ~^9.  A  .  ~_p        (2) 

[(2).  Def.  Disjunc.  Implica.]         I-.  theorem. 

This  is  Mr  Russell's  primitive  proposition  *1.2  given  above. 

Theorem  12 

h:  q.'^.pwq. 

Dem. 

Two  cases  need  only  be  treated  : 

I'' :  q  true.  Then  (Th.  3)  ~  q  is  false.  Hence  (Th.  6) 
'  ~  jj  A  <^  q  '  is  true.  Hence  ~  (~  ^j  A  ~  q)  is  false,  by  Th.  3. 
Hence 

[Ax.  3]  h :  (/ .  A  .  <^  ( ~  p  A  ~  (/)  ( 1) 

2° :  q  false. 

[Th.  r,  |-~(~pA~g)j  F.  , .  A  .  ~  ( ~  ,,  A  ~  ,/)  (2) 

[(1).  (2).  Def.  Disjunc.  Implica.]      h.  theorem. 

This  is  Mr  Russell's  primitive  proposition  *l.o  given  above. 

Theorem  13 
h:  pv  q  .D  .  qy  P' 

Dem. 

[Th.  7]  h :  '  ~  jt)  A  ~  g '  and  '  ~  g  A  ~  p  ' 

of  the  same  truth- value  (1) 

[(1).  Th.  3.  Ax.  3]  h:  ~  jj  A  ~  (/ .  A  .  ~  ( ~  ry  A  ~  p)  (2) 

[(2).  Def  Disjunc.  Implica.]     h:  theorem. 

This  is  Mr  Russell's  primitive  proposition  *1.4  given  above. 

Theorem  14 
V :  p  y  {q y  r) ."^  . qy  {p M r). 

Dem. 

[Th.  9]     I-:  '~p  A  ~(~(/ A  ~?-)' a-nd '~(/ A  ~(~|)  A  ~?-)' 
of  the  same  truth-value  (1) 

[(1).  Th.  3.  Ax.  3]  h:   ~  ;j  A  ~  (~  (?  A  ~  /•) 

.  A  .  ~  [~  q  A  '^  {r^  p  A  ~  r)]  (2) 

[(2).  Def.  Disjunc.  Implica.]     h:  theorem. 

This  is   Mr  Russell's  primitive  proposition  *1.5  given  abt)ve. 


30  Mr    Van   Horn,  An   Aj'iunt  in  Sfjtnbolic  Logic 

Theorem  15 


J 


\-:.qDr.D:pvq.D.2i'v  r. 

Bern. 

There  are  three  cases  to  be  discussed : 
1" :  li  p  is  true,  or  if  r  is  true,  or  if  both  p  and  r  are  true, 
q  being  any  elementary  proposition. 

[Th.  8]  }-:   r^l.  A.^i^p  A  '^r)  (1) 

[(1).  ::ii.  Th.  10]  h:  Z.  A  .~(-2J  A  ~?-)  (2) 

[(2).   ~i^^  ~5j  h:   ~p  A  ~(/.  A  .  ~(~2)  A  ~  r)(3) 

[(3).  Th.  3.  Th.  6] 

h:  q  A  ^  r .  A  .  <^  [^  p  A  '^  q .  A  .  <^  (r^  p  A  '^  r)]  (4) 

Taken  together  with  the  Definitions  of  Implication  and 
Disjunction,  (4)  gives  the  theorem  in  this  case. 

2° :  If  both  p  and  r  are  false,  but  q  true.  In  this  case  ~  ^j  and 
oo  r  are  true  by  Th.  3.  Hence  (Ax.  S)  '  ^  p  A  ~  r '  is  false.  The 
proof  in  this  case  proceeds  as  folloAvs : 

[Th.  3]  1-:   ~(-2)  A  ~  r)  (5) 

Since  q  is  true,  '^  q  is  false  (Th.  3). 

[Th.  6]  h.   - 19  A  ~  5  (6) 

[(5).  (6).  Th.  3.  Ax.  3]  V:   ^[^^p  A  ^q.  A.'^i^p  A  ^r)]{1) 

By  Ax.  S,  '  q  A  <^  ?' '  is  in  this  case  false. 

[(7).  Ax.  3] 

h:  q  A  ~r.A.'^[~/jA  <^5.A.~(~jjA  ~  ?•)]     (8) 

As  in  the  previous  case  this  result  gives  the  theorem. 

3° :  All  three  false.  Hence  ~  p  and  ~  ?■  true  as  before.  In 
this  case  '  ^  p  A  <-^  q'  is  false  by  Ax.  3.  The  proof  in  this  last 
case  proceeds  thus : 

[Th.  3,  as  in  2°]  h.   ~(~pA~?')  (9) 

[(9).  Ax.  3]  h:   ~p  A  ~  g.  A  .  ~(~p  A  ~  r)  (10) 

In  this  case  q  and  f--'  r  are  of  opposite  truth- values. 

[Ax.  3]  h:  f^  A~r  (11) 

[(10).  Th.  3.  (11).  Ax.  3] 

h:  ^A~?'.A.'^[~pA'^^.A.'^  (~i^  ^  ~  ''')]     (12) 

As  in  the  two  preceding  cases,  this  result,  together  with  the 
Definitions  of  Implication  and  Disjunction,  gives  the  theorem. 

No  other  cases  can  arise.     Hence  the  theorem. 

This  is  Mr  Russell's  primitive  proposition  *1.6  given  above. 
It  asserts  that  an  alternative  may  be  added  to  both  premise  and 


Mr   Vail    Horn,  An  Axiom   in   Si/nihohc  Loffic  ol 

conclusion  in  any  implication  without  impairing  the  truth  of  the 
implication. 

This  completes  the  list  of  Mr  Russell's  primitive  propositions 
that  I  proposed  for  proof  by  means  of  my  axiom,  on  the  basis  of 
the  definitions  given  in  this  paper  of  the  four  fundamental 
operations  of  logic. 

I  now  propose  to  prove  two  propositions  which  can  take  the 
place  of  his  definitions  of  Implication f  and  Conjunction  j,  or  Joint 
Assertion. 

Theorem  16 

Dem. 

[Th.  4^^'"'^]  h:  p  A  -</.D.;9  A  ~g  (1) 

[Th.  10]  D .  ~  (~  p)  A  ~  ry         (2) 

[(2).  Def.  Implica.  Disjunc]  h:  pD  q  .D  .  ^^  pw  q             (3) 

[(1).  Th.  10]                            h:  ~(~j9)A  ~(y.D.j)A  ~  7  (4) 

[(4).  Def.  Implica,  Disjunc]  h :   '^py/q.D.pDq             (5) 

[(3).  (5).  Def.  Equiv.]  h  :  theorem. 

Theorem  17 

h:  p  .  q  .  =  .  ^  { '^  p  W  ^  q). 
Dem. 

[Th.  4  *"  ^^  ^  '^^]         h:   ~(jo  Ary).D.~(p  A  (/)  (1) 

[Th.  10]  D.~[~(~y/)  A  ~(~^)](2) 

[(2).  Def.  Conjunc.  Disjunc]        I- :  p .  q .  D  .  ^^  (^  p  v  r^  q)  {S) 
[(1).  Th.  10]  h:    ~[~(~p)A  ~(~5)].D.~(p  Afy)(4) 

[(4).  Def.  Conjunc.  Disjunc]        h  :   '^  (^  p  v  ^  q) .  "D  .  p  .q  (5) 
[(3).  (5).  Def.  Equiv.]  h :  theorem. 

With  these  theorems  established  the  development  of  the 
Principia  Mathematica  can  proceed  as  given  by  its  authors. 
All  that  I  have  done  is  to  reduce  the  number  of  axioms  needed 
for  that  development. 

Baptist  College, 
Rangoon,  Burma. 

t  Op.  cit.  Vol.  I.  p.  98,  *1.01.  +  Ibid.  p.  116,  *3.01. 


32  Mr  JSicod,  A  Reduction  in  the  nimiber 


A  Reduction  in  the  number  of  the  Primitive  Propositions  of 
Logic.  By  J.  G.  P.  NicOD,  Trinity  College.  (Communicated  by 
Mr  G.  H.  Hardy.) 

[^Received  and  read  80  October  1916.] 

Of  the  four  elementary  truth-functions  needed  in  logic,  only 
two  are  taken  as  indefinables  in  Principia  Mathematica.  These 
two  have  now  been  defined  by  Mr  Shefferf  in  terms  of  a  single 
new  function  p  |  q,  " p  stroke  q."  I  propose  to  make  use  of  Mr 
Sheffer's  discovery  in  order  to  reduce  the  number  of  the  primitive 
propositions  needed  for  the  logical  calculus. 

There  are  two  slightly  different  forms  of  the  new  indefinable, 
for  we  may  treat  2:)\q  as  meaning  the  same  thing  as  either 
~jj  .  ~g,  or  <^p}/  ^qt-  The  definition  of  <^p  is  the  same  in 
both  cases,  namely  p  \  p,  while  that  of  pv  q  simply  changes  from 
p/q  \p/q  with  the  AND-form  into  p/p  \  qjq  with  the  07^-form. 

However,  the  best  course  is  for  us  to  define  all  the  four  truth- 
functions  directly  in  terms  of  the  new  one.  In  so  doing,  we  find 
that,  while  the  definition  of  ~j9  remains  the  same,  and  those  of 
pv  q,  p  .  q  simply  permute,  as  we  pass  from  the  ^iV^D-form  to  the 
Oi^-form,  the  definition  of  pO q  is  simpler  in  the  latter  form.  It 
is  p  I  qjq,  as  against  j;/j)  j  q  \p/p  \  q. 

The  OJ?-form  is  therefore  to  be  preferred  §. 

Definitions. 

f^p  .  =  . p\p         Df.  pvq.^.plpiq/q     Df. 

pO  q  .  =  .  p\  qjq     Df.  p  .  q  .  —  .  p/q  I  p/q     Df 

Remaeks  on  these  Definitions, 

One  ought  not  to  aim  at  retaining  before  one's  mind  the 
complex  translation  into  the  usual  system,  "-^pv^q"  as  the 
"real  meaning"  of  the  stroke.  For  the  stroke,  in  the  stroke- 
system,  is  simpler  than  either  ~  or  v,  and  fi-om  it  both  of  them 
arise.  We  may  not  be  able  to  think  otherwise  than  in  terms  of 
the  four  usual  functions ;  it  will  then  be  more  in  accordance  with 
the  nature  of  the  new  system  to  think  of  the  j ,  not  as  some  fixed 
compound  of  -^  and  v,  but  as  a  bare  structure,  out  of  which,  in 
various  ways,  ~  and  v  will  grow. 

+  Sheffer,  Trans.  Amer.  Math.  Soc.  Vol.  xiv.  pp.  481—488. 
X  Sheffer,  loc.  cit.,  footnote  f,  p.  488. 

%  p\q  thus  corresponds  to  what  is  termed  the  Disjunctive  relation  in  Mr  W.  E. 
Johnson's  writincrs. 


|i 


of  the  Primitive  Propositions  of  Logic  33 

The  above  definitions  give  clear  expression  to  the  symmetiy 
between  OR  and  AND  ;  and  this,  notwithstanding  the  choice  that 
we  had  to  make  between  an  Oi?-forni,  and  an  AND-iorva.  This 
is  of  some  interest,  because,  in  general,  the  very  symmetry  forces 
upon  us  an  arbitrary  choice,  which,  in  turn,  quite  obscures  the 
symmetry. 

I  shall  use  q  for  q\q  whenever  convenient.  Observe  that 
p  I  q,   i.e.  pD  q,    forms   a   natural   symbol    |  for   implication, 

allowing  of  permutation  ~q\  p.  We  may  notice  in  general  that 
the  new  system  brings  the  four  functions  into  relations  far  closer 
than  those  in  Mr  Russell's  system.     For  instance,  in 

p/p\p/p-\.p/p 
the  two  propositions  pv  p  .D  .  p  and   r^pv  p  coincide. 

Every  stroke-formula  falls  into  two  parts  on  the  right  and  left 
of  a  central  stem.  It  will,  therefore,  add  to  clearness  to  use  black 
type  instead  of  dots  to  indicate  the  central  symbol.  Further, 
slanting  strokes  are  covered  by  straight  ones  :  thus  p/q  j  p/q  stands 
for  (p\q)\  (pj  q). 

The  definition  of  the  two  primitive  notions  of  the  Principia 
in  terms  of  a  single  new  one  tends  to  reduce  the  number  of  the 
primitive  propositions  needed.  But  how  far  does  this  reduction 
actually  occur  ?  Does  it  extend  beyond  the  obvious  substitution 
of  "  If  p  and  q  are  elementary  propositions,  p\q  is  an  elementary 
prop."  (Sheffer,  p.  488)  for  *r7  and  *1'71,  stating  the  same  for 
~  p  and  py  q  respectively  ?  The  reduction  goes,  as  we  shall 
presently  find,  very  much  farther. 

It  has  first  to  be  said,  in  order  that  we  may  be  as  precise  as 
possible,  that  the  tuhole  amount  gained  in  applying  the  stroke- 
definitions  cannot  with  complete  certainty  be  attributed  to  them. 
For  Mr  Russell's  system,  as  it  now  stands,  has  not  said  its  last 
word  in  that  matter. 

Incidentally,  I  found  that  *1'4,  pv  q  .D  .  q  y  p,  can  be  proved 
by  means  of  the  other  four,  with  the  unimportant  change  of  *1'3, 
q  .  "^  .  pv  q  into  q  .  "^  .  q  v  p.    In  "Association,"  *1*5,  writers  for  r  : 

p  y  {q  y p)  .  1^  .  qv ij)  V p). 

The  left-hand  side,  by  the  help  of  q  ."D  .  qvp  and  "  Summation," 
will  be  found  to  be  implied  in  pv  q.  The  right-hand  side,  like- 
wise, hy  p  V  p  .  D  r  p,  and  "  Summation,"  will  be  found  to  imply 
qvj).     The  result  then  follows  by  using  "Syllogism"  (obtained 

from  "  Summation  "  with  the  transformation  — -  f)  twice. 

p 

P        V    p' 
t  By  -  or  ^-^  I  mean  (following  Mr  Russell)  the  substitution  of  p  for  q  or 

p,  p'  for  q,  q'.     By  {e.g.)  P~  I  mean  the  result  of  effecting  the  substitution  in  P. 
VOL.   XIX.   PT.   I,  3 


34  Mr  Nicod,  A  Reduction  in  the  number 

Let  us,  however,  take  Mr  Russell's  eight  propositions  in  the 
form  given  in  Principia.  It  is  my  object  to  reduce  them  to  three 
— two  non-formal  and  one  formal — by  means  of  the  stroke-defi- 
nitions given  above. 

It  can  be  shown,  as  a  first  stage,  that  two  formal  propositions 
are  enough,  namely : 

(1)  p\l)/p. 

(2)  p\q/q\s/q\^. 

The  first  proposition  is  the  form  of  "  Identity "  (p  D  p)  in  the 
stroke-system.  It  would,  at  first  sight,  appear  more  natural  to 
adopt  the  order  q/s  \  p/s  in  the  left-hand  side  of  (2),  since 

p\qlq-'^-qls\p/s 

is  the  syllogistic  principle  of  the  stroke-system,  giving  "  Syllogism," 
pD  q  .D  :  q  D  s  .  D  .pDs  when  s  |  s  is  written  for  s. 

It  will  however  be  found  that  the  inverted  order,  s/q  1  p/s,  is 
much  more  advantageous  than  the  normal  syllogistic  order, 
q/s  \p/s.  For,  owing  to  this  "  twist,"  Identity  and  (2)  yield 
"  Permutation,"  s/p  \  p/s,  which  now  enables  us  to  eliminate  the 
twist  in  (2),  and  revert  to  the  normal  order.  From  the  three 
propositions  thus  obtained,  the  rest  follow. 

This,  by  the  way,  illustrates  the  following  fundamental  fact. 
Which  form  of  a  given  principle  is  the  most  general,  and  contains 
the  maximum  assertion,  is  a  function  of  the  symbolic  system  used. 
Thus,  for  instance,  in  Mr  Russell's  system, 

p  .D  .  qwp         (a) 
is  more  general  than         p  .0  .  qD  p        (b) 

since  (h)  is  (a)  with  <^q  for  q.  In  the  stroke-system,  on  the 
contrary,  p  \  q/q  \  p/p,  meaning  the  same  thing  as  (a),  is  less  general 

than  p\  q  \p/p,  whose  meaning  is  that  of  (b),  since  it  is  obtained 
from  it  by  writing  q\q  for  q. 

A  further  step  has  to  be  made  in  order  to  be  left  with  only  one 
formal  primitive  proposition.  It  consists  in  adapting  to  better 
advantage  the  form  of  the  primitive  propositions  to  the  properties 
of  the  stroke-symbolism  where  implication  is  concerned.  We  had 
above 

p'^q  .  =  .p\  q/q    Df 

If  we  look  for  the  meaning  of  the  general  form  p  \  r/q,  we  find  this 
to  be  oo  29  V  ~  (~  r  V  ~  5'),  i.e.  p  .D  .  r  .q.  We  thus  come  to  the 
fundamental  property  that,  in  the  new  system,  p"^  q  is  a  case  of 
p  .D  .  s  .  q,  whereas  in  Principia  the  contrary  relation  of  course 
holds, 


i 


of  the  Primitive  Propositions  of  Logic  35 

This  leads  us  to  substitute  p  \  r/q  for  'p  \  q/q  in  the  "  left-hand 
sides  "  of  both  the  non-formal  rule  of  implication  and  the  syllo- 
gistic proposition  (2)  above.  The  reform  may  be  further  extended 
to  the  proposition  (2)  as  a  whole,  which  might  be  given  the  form 
P  !  S/Q  instead  of  P  \  Q/Q,  with  the  proviso,  if  the  proposition  is  to 
remain  true,  that  *S'  must  be  implied  in  P.  Now,  for  S,  write  the 
pioposition  (1)  above,  p\p/p  ;  for  (as  we  at  this  early  stage  know 
"  unofficially  ")  a  true  proposition  will  be  implied  by  everything. 

We  then  have  the  three  primitive  propositions  of  the  stroke- 
system  : 

(  I.  If  p  is  an  elementary  proposition,  and  q  is  an 
Non-  elementary  proposition,  then  p\q  is  an  elementary  pro- 
formal  1  position  f. 

\   II.     If  J)  [  r/q  is  true,  and  p  is  true,  then  q  is  true. 

This  is  the  non-formal  rule  of  implication,  *1'1,  with  the  modifi- 
cation just  explained. 

Formal     III.  p  j  q/r  \t\t/t.\.  s/q  [p/s. 

I  shall  call  II  "  the  Rule,"  and  III  "  the  Prop." 

Remarks  on  these  Primitive  Propositions. 

Observe  p  r/q  in  II,  while  p  |  q/r  in  III.  This  alternance  will 
prove  essential  for  the  working  of  the  calculus.  

In  III,  I  shall  use  ir  for  1 1  t/t,  P  for  p  j  q/r,  Q  for  s/q  \p/s,  and 
shall  speak  of  III  as  P  \  tt/Q. 

P  I  ir/Q,  by  the  Rule,  yields  the  same  result  as  the  syllogistic 
proposition  (2)  above,  when  the  left-hand  side  P  is  a  truth  of 
logic.  This  restriction  of  the  syllogistic  form  to  its  categorical 
use  with  an  asserted  premiss  is  a  peculiar  character  of  the  first 
proofs  to  follow,  and  is  of  some  philosophical  interest. 

One  feels  inclined  to  think  that  III  merely  asserts  together 
(1)  and  (2)  above.  This  view,  whatever  may  be  the  amount  of 
truth  it  contains,  takes  AND  too  much  as  a  matter  of  course, 
and  tends  to  lose  sight  of  (a)  the  fact  that  III,  as  a  structui;^^s 
simpler  than  (2)  alone  :  for  III  is  (2)  with  t  \  t/t  instead  of  s/q  \p/s ; 
and  (y8)  the  very  real  step  from  p  .q  to  q,  together  with  the  philo- 
sophical difference  between  two  assertions  and  only  one. 

The  main  steps  in  the  formal  deduction  are : 

1.  Proof  of  "  Identity,"  t  \  t/t. 

2.  Passage  from  P  \  ir/Q  to  the  usual  implicative  form  P  [  Q/Q. 

3.  Elimination  of  the  twist  s/q  \p/s  in  Q,  and  return  to  the 
normal  order  q/s  \p/s. 

t  This  is  the  proposition  shown  by  Sheffer  to  imply  the  analogous  propositions 
*1*7  and  *1-71  in  Principia. 

3—2 


36  Mr  Nicod,  A  Reduction  in  the  numher 

4.  Proof  of  "  Association,"  p  \  q/r  .D.q  [p/s. 

5.  Theorems  equivalent  to  the  definitions  of  p  .  q,  p  0  q  in 
Principia. 

Proof  of  Identity,  t\t\t. 

As  this  first  proof  from  a  single  formal  premiss  stands  in  a 
unique  position,  I  shall,  without  in  any  way  obscuring  the  precise 
play  of  the  symbols,  expound  it  after  a  more  heuristic  order  than 
is  usually  followed. 

We  start  with  the  Prop.  P  |  tt  |  Q,  and  the  Rule  enabling  us  to 
pass  from  the  truth  of  P  to  that  of  Q ;  and  we  have  to  prove  tt. 
This  can  only  be  reached  through  some  proposition  of  the  form 
-4  1 5 1  TT,  where  A  is  a  truth  of  logic f.  The  proof  will  thus  consist 
in  passing  from  P  |  tt  |  Q  to  J.  1 5  |  tt  by  some  permutative  process. 

A  simple  two-terms  permutative  law  s  1 5'  |  ^  |  5,  we  do  not  yet 
possess.  Our  Prop,  yields  only  a  roundabout  three-terms  per- 
mutation, slglpjs,  subject  to  the  condition  of  ^jglr  being  a 
truth  of  logic  f.     This,  however,  is  enough  for  our  purpose. 

In  the  Prop.,  write  t.  for  p,  q,  r : 

(a)  7r|7r!Qi, 

Qi  being  s|^|^|s.     Write  now  tt  for  p,  q;  Q^  for  r:  then  by  (a) 
and  the  Rule, 

(b)  S  !  TT  I  TT  I  s. 

From  (b),  in  the  same  manner. 


(c)  u  I  tt/s  I  s/tt  j  u. 

This  enables  us  to  pass,  by  the  Rule,  from  P  |  tt  |  Q  to 

(d)  Q|7r|P. 

In  order  to  complete  the  proof  of  tt,  we  need  only  find  some 
expression  which :  (a)  can  be  a  value  for  P,  i.e.  is  a  case  of  p\q\  r, 
and  (/3)  is  implied  in  some  truth  of  logic,  say  T.  For,  by  T'lP  |  P, 
the  Prop.,  and  the  Rule,  as  above, 

(e)  s\P[T\~s. 

In  (e),  write  Q  |  tt  for  s:  first  by  (d)  and  the  Rule,  then  by  T 
and  the  Rule,  we  obtain  T\Q\7r,  and  so 

(/) 

t  This  use  of  the  Rule  by  anticipation,  with  still  undetermined  P's  and  Q's,  is 
in  truth  contrary  to  the  nature  of  a  non-formal  rule,  which  must  never  be  used  to 
build  up  the  structure  of  an  argument.  It  must  always  be  possible  to  dispense 
with  all  such  '  anticipated '  assertions  in  the  final  form  of  a  proof.  This  will  be 
seen  to  be  very  easy  in  the  present  case. 


of  the  Primitive  Propositions  of  Logic  37 

Now,  Qi  I  7r|  TT  fulfils  (a)  and  {^).  For  (a)  tt  being  the  complex 
expression  t\t\t,  is  a  case  of  the  form  q  \  r,  and  (/3)  we  have,  by 
(c)  above,  tt  i  tt/Qi  \  Qi/tt  \  tt,  and  by  (a)  tt  |  tt  |  Qj. 

To  obtain  the  strictest  development  of  the  proof  we  have  only 
to  write  Qi/tt  tt  for  P  and  ir  ;  tt/Qi  for  T  all  through  the  preceding 
argument. 

Permutation,                 s  |  p  |  p  I  s 
Gives  sv  p  .1)  .  py  s  hy  ^ , 

Dem. :   Prop.    - — — — -  ,  Id.,  and  Rule. 
j3     q     r 

Tautology,  p/p  \  p/p  \  pjp 

i.e.    py p  .  0  -p 

Dem.:  Id.^,  Perm.,  and  Rule. 
P 


Addition,  s\p\sls 

Gives  s  ."D  .py  s  by  —   . 
Dem. :  By  Perm,  (twice),  p  \  s/s\sjs  \p         (a) 


By   Prop,   ^-y qrs '   ^  (")'   ^  W.  +,   p  \  s/s  \  s 

By  Perm.,  result. 

Return  froivi  Generalised  Implication  P  \  tt/Q  to  P  Q/Q. 


Lemma,  pjp  \  s/j) 

Dem. :  By  Perm,  (twice),  s/p  \  p/s         (a) 

By  Prop.  -^ — ,  I-  {a), 

-^         ^     p      q     r     s 


u\p\  s/p  I  It 
Write  p/p  for  ii :  by  Id.  and  Perm,  (twice),  result. 

t  \-  (a)  means  the  use  of  the  Rule  to  pass  from  a  to  b  iu  a    sjl). 


38  Mr  Nicod,  A  Reduction  in  the  number 


Theorem,  P\irlQ\QIQ\P 

Dem. :  Prop.  -^^^-^^ -^ ,  r  Lemma,  reeult. 

p       q,  r     s 

Hence,  by  Perm.,  P  \  Q/Q,  i.e. 

P I  5'/^  I  s/q  i  P/^         (^') 


Syllogism,  i?  |  5*/^  I  q/s  \  p/s 

o     s  s 
Gives  _p  D  (/ .  D  :  fy  D  s  .  D  .  jj  D  6'   for  ^^ — 

Dem. :  In  this  Dem.,  Permutation  is  used  to  correct  the 
twisting  action  of  S\  much  as  handwriting  has  first  to  be  inverted, 
if  it  is  to  be  seen  right  in  a  mirror. 

By  8'  ~ -^ ,  I"  Perm.,  and  Perm., 

•^         p     q,  r     s 

qjs  I  u  I  u  I  sjq  {a) 

•^  p  q,  r  s 


qjs  I  u  I  sjq  I  u         (b) 
By  ^-i^lgA^     ^/glW^    g/HW^^  H^',  h6,  result. 


Association,  p  \  q/r  |  q  \pjr 

The  structure  of  the  proof  is  this  : 

Syll."     Il''     '• 
p    q,  r    s 

gives  _p  I  g/r  .  D  :  q/r  |  ?' .  {p/r. 


We  now  need   only  the   Lemma  q  \  q/r  |  r  for  our  result  to 
follow  by  Syll.  twice. 


Lemma,  q  \  q/p  |  p 

The  proof  of  this  lemma — call  it  L — is  as  follows  :  We  prove 
(a)  q  I  LjL,  (b)  L/L  \  q/q.  From  this,  by  Syll.  and  TautoL,  the 
result  follows. 

Dem. :  (a)  By  Syll.  ^  , 
r,  s 

p\qlq-:^-q/p\plp  (1) 


of  the  Privative  Propositions  of  Logic  89 

By  Ackl,  SylL,  I-  (1), 


q.  D:q/p\p/2) 

(2) 

The  right  side  of  (2)  implies,  by  Syll., 

plp\p.'^.q/p\p 

(3) 

By  Id.,  Perm.,  Add.^/^'^'  -'^ , 
^        '              '                 p,          q' 

q.D:p/p\p 

(4) 

By  Syll.  twice,  h  (2),  h  (3),  h  (4), 

qD  :  q  ."^  .  q/p  \p,     i.e.  q 

L/L. 

(b)     By  lemma  to  Syll.,  q/q\s/q;  by  Perm,  and  Syll,  q/q  Iq/s. 
Hence,  q/q  \  L/L  ;  by  Perm.,  L/L  \  q/q. 
Now,  by  Syll. : 

L/L  1  q/q  .D:q\L/L.D  .  L/L  |  L/L. 

By  1-6,  h  a,  and  Taut.  -,  result.     We  can  now  complete  the  proof 
of  '  Association.' 


Association,  p  \  q/r  \  q  \  p/r 


Dem. :  By  Syll.,     /)  |  q/r  .  D  :  q/r  \r  .\.  p/r 
By  Syll.  twice,  h  Lemma,  result. 

Summation,  qDr  .D  :  pvq  .D  .pv  r 

Dem. :  By  Syll.,  Assoc, 

q  \s  .D  :  p\  q/r  .  D  .  p\s         (1) 

^         .       s/s,     q,     p/p  , 

By  (1) — ^-^,  result. 

-^  s,      r,      p 

Theorems  Equivalent  to  the  Definitions  of  p  Dq,  p  .  q, 
IN  Principia. 

p"^  q  •  3  .  ^pv  q,  and  reciprocal  theorem. 
That  is,  p  I  q/q  .  D  .p/p  \  q/q. 

Bern. :  Taut.,  and  Syll. 

sis        D 

Reciprocal  theorem  by  Add.  -^ — —  ,  and  Syll. 

.9,      p 

p\  q  ."D  .  ^p  V  ~  q,  and  reciprocal  theorem. 
That  is,  j9 1  5  .  D  .  p/p  \  q/q. 


40  Mr  Nicod,  A  Reduction  in  the  nmnher 

Devi. :  Taut.  SylL;  then,  Perm.,  Taut.,  and  SylL,  or  S'. 

Reciprocal  theorem  by  Add,  ^-^  instead  of  Taut. 

^  .  g  .  D  .  ~  {"^p  V  '^q)  and  reciprocal  theorem. 
That  is,  p  .q  .D  .  p/q  \  p/q. 

Dem. :  Id,,  Def,  of  ~,  preceding  theorem,  and  Syll. 
Reciprocal  theorem  in  the  same  manner. 


Appendix, 

After  the  substance  of  this  paper  had  been  written,  I  was 
given  the  opportunity  of  seeing  Mr  Van  Horn's  very  interesting 
and  original  paper  dealing  with  what  is  practically  the  same 
subject,  Mr  Van  Horn  recognises  clearly  the  superiority  of  what 
has  been  called  above  the  Oi^-form  over  the  j4iVD-form  chosen 
in  Sheffer's  text.  This  deserves  the  more  notice,  as  Mr  Van 
Horn,  I  understand,  had  not  Sheffer's  article  at  hand  in  the  time 
he  was  writing  his  own  paper.  His  A,  as  will  be  seen  from  the 
definitions  he  gives,  is  indistinguishable  from  |.  I  was  much 
attracted  by  the  harmonious  character  of  Mr  Van  Horn's  third 
Axiom.  It  seems  to  me  therefore  all  the  more  desirable  that 
certain  objections,  which  Mr  Van  Horn's  proofs  in  their  present 
form  naturally  suggest  to  the  reader,  should  be  dealt  M'ith, 

(a)  It  is  not  quite  plain  to  me  whether  "  of  the  same  truth- 
value  "  (say  S  for  short),  "  of  opposite  truth-values  "  (say  0),  are 
used  as  indefinables,  or  as  abbreviations.  If  the  former,  we  have 
no  right  to  go,  e.g.,  from  p  0  q,  and  '^p,  to  q,  etc.,  without  some 
axiom  to  that  effect,  connecting  0  and  S  with  A,  If,  on  the 
other  hand,  S  and  0  are  abbreviations — as  it  seems  to  me  they 
are — the  two  parts  of  Axiom  3  stand  for  not  less  than  four 
propositions : 

1,  If  jj  and  q,  '^{pAq). 

2.  If  (^p  and   ~(/,       pAq. 

3.  If  p  and  ^q,  pAq. 

4,  If  ^p  and  q,  pAq. 

We  cannot  assert  the  first  two,  or  the  last  two,  or  all  four, 
propositions  together,  because  we  should  then  need  p  .  q  .  D  .  p, 
p  .  q  .  D  .  q,  before  we  could  make  any  use  of  such  a  synthetic 
Axiom, 


of  the  Primitive  P7'opositions  of  Logic  41 

This  uncertainty  as  to  the  status  of  S  and  0  is  not  without  its 
effect  upoii  the  proofs.  Consider,  for  instance,  Th.  3.  In  the  proof, 
"1°:  p  true.  By  Axiom  3,  pAp  false"  will  be  seen  to  require  p  Sp, 
concerning  the  origin  of  which,  and  the  relation  it  has  to  p  D  j^ 
(Th.  4),  which  it  indirectly  serves  to  prove,  Mr  Van  Horn  says 
nothing. 

(/3)  In  his  extensive  use  of  the  Principle  of  Excluded  Middle, 
Mr  Van  Horn  makes  no  explicit  mention  of  the  last  steps,  that 
lead  from  pOq,  ^^  pDq,  to  q.  These  steps  would  seem  to  require 
several  propositions:  (1)  those  carrying  us  from  ^^pyp  to  qvq 
— "  Summation,"  plus  "  Permutation,"  presumably — and  (2)  "  Tau- 
tology "  qv  q  .D  .  q.  As  Mr  Van  Horn  uses  the  principle  of 
Excluded  Middle  in  this  particular  way  in  the  first  formal  proof 
given — that  of  Th.  3 — both  the  principle  itself  and  the  proposi- 
tions required  for  its  use  ought,  I  think,  to  be  deduced  immediately 
from  Axiom  3 ;  and  I  do  not  see  how  this  is  possible. 


42  Mr   Watson,  Bessel  functions 


Bessel  functions   of  equal   order   and   argument.     By  G.  N. 
Watson,  M.A.,  Trinity  College. 

[Received  1  November  1916:   read  13  November  1916.] 


A  proof  of  the  approximate  formula 


Jn{n)' 


TT  2»  3«  w« 


(the  order  and  argument  of  the  Bessel  function  being  equal  and 
large)  was  apparently  first  published  by  Graf  and  Gubler*, 
although  the  formula  had  been  stated  by  Cauchyf  many  years 
before.  The  formula  has  been  discussed  more  recently  by 
Nicholson  J  and  by  Lord  Rayleigh§,  while  Debye||  has  given  a 
complete  asymptotic  expansion  of  Jn{n)  in  descending  powers 
of  71 ;  this  expansion  is  obtained  by  the  aid  of  the  elaborate  and 
powerful  machinery  which  is  provided  by  the  mode  of  contour 
integration  known  as  the  "Methode  der  Sattelpunkteir"(Methode 
du  Col,  method  of  steepest  descents). 

The  earlier  writers,  just  mentioned,  employed  Bessel's  formula 


1  /■'" 
Jn  (*')  =  —  I   COS  (nO  —  X  sin  6)  dd, 
ttJo 


valid  when  n  is  an  integer,  and  it  is  by  no  mea.ns  obvious  to  what 
extent  their  methods  of  approximating  are  valid**. 

As  the  correctness  of  the  approximation  can  be  established 
without  the  use  of  contour  integration  on  the  one  hand  and 
without  appealing  to  physical  arguments  ff  on  the  other  hand, 
it  seems  to  be  worth  while  to  write  out  a  formal  and  rigorous 
proof  (based  on  comparatively  elementary  reasoning)  that,  when 
n  is  large  and  real,  then 

*  Einleitung  in  die  Theorie  der  Bessehcheii  Funktionen,  i.  (1898),  pp.  96 — 107. 

+  Comptes  Rendus,  xxxviii.  (1854),  p.  993;   Oeuvres  (1),  xii.  p.  163. 

J  Phil.  Mag.,  August  1908,  pp.  273—279. 

§  Phil.  Blag.,  December  1910,  pp.  1001—1004. 

II  Mathematische  Annalen,  lxvii.  (1909),  pp.  535 — 538. 

1[  This  method  of  discussing  Je"/W<^(s)(fi;  consists  in  choosing  a  contour  on 
which  If{s)  is  constant,  and  so  Bf{s)  falls  away  from  its  maximum  as  rapidly  as 
possible  (/(s)  being  monogenic);  it  is  to  be  traced  to  a  posthumous  paper  by 
Eiemann,  Werke,  1876,  p.  405. 

**  See  §  4  below. 

ft  For  example  Kelvin's  "Principle  of  stationary  phase"  {Phil.  Mag.,  March 
1887,  pp.  252—255 ;  Math.  Paiyers,  iv.  pp.  303—306)  is  really  based  on  the  theory 
of  interference.  See  also  Stokes,  Camh.  Phil.  Trans,  ix.  (1850),  p.  175,  foot-note 
(Math.  Papers,  ii.  p.  341). 


of  equal  07'der  and  argument 


43 


Jnin)=^ 


7r2*3-^ 


2.     In  order  not  to  restrict  ourselves  to  the  case  in  which  n 
is  a  positive  integer,  we  take  the  Bessel-Schlafli  integral*,  namely 


sin  IITT 


J II  (^)  =  -      cos  {nd  —  X  sin  6)  dO  — 


,-nd  -X  sinh  ^ 


dO, 


(which  is  valid  whether  n  be  an  integer  or  not),  and,  after  writing 
n  for  x,  we  integrate  by  parts.     This  process  gives 


Jn  (it)  =  ^ 


d 


nir  j  0  1  ""  cos 

sin  mr 


-^  -Tj.  {sin  n  (0  —  sin  6)]  dd 


+ 


d 


IT     J  0  1  +  cosh  9  dd 


|g-«(^  +  siuh^),    ,^ 


nir 


sinw(^  — sin^) 


1  —  cos  6 
1 


+ 


sm  nir 


-n{0-^sm}\d)-\  ^ 


1  +  cosh^ 


+ 


'^smn(^-sm^)^^^^^^^ 


/, 


mr]  Q     (1  —  cos  6)" 

sinnTTp       sinh^      ^-n(d  +  s.\nhe)  ^n 

"^        TT        Jo     (T+COsh^)^^  ;''^' 

The  integrated  parts  cancel ;  and 

^^^^^         -n[6  +  ^mh0)^0  ^  f"(l+C0sh6^)«-»(^  +  «i"l^^)(/^ 
(1  +  cosh^)-  Jo 

=  lln; 

and  so,  when  ?«  is  large  and  real, 

r  y  .       1   f ''  sin  ^  sin  w<f>  , ,       „  ,     ^, 

?i7r.lo  (1  —cose')*     ^ 

where  </>  has  been  written  in  place  of  6  —  sin  6.     It  is  obvious  that 
<l>  inci-eases  steadily  from  0  to  tt  as  ^  increases  from  0  to  tt. 

When  6  is  small,  (f)r^^6^  and  sin  ^ .  (1  — cos  0)~^  ^  80-'^.   Hence, 
as  ^  ->  0,  • 

<^^  sin  0         8 


(1  —  cos  0y      Qi  ' 
Now  write    |  (60)*  sin  ^ .  (1  -  cos  0)~'  =/  (</>) ; 


Schlatii,  3Iath.  Ann.  m.  (1871),  p.  14«. 


44  Ml'   Watson,  Bessel  functions 

then  it  is  fairly  evident*  that  when  0  ^0  ^ir  (i.e.  when  0  ^  0  ^  tt), 
fi{<f>)  is  bounded  and  has  only  a  finite  number  of  maxima  and 
minima  (and  therefore  it  has  limited  total    fluctuation).     Con- 


sequently, since f       -^jr    ■•  sim/rc/i/r  is  convergent,  we  have;]: 
.  0 
v  77  r  ^ 

Lim  n~i  ^    (f)"'^ sin  (n(f>) .  f\  {(f))  d(fi  =/i  (0)  |     yjr'"  sin  yjr d-yjr. 

Therefore,  since /i(0)=  1,  we  have 

n-ijy~fsm(ncf)).f,{cf>)d<f>  =  ^T(V>  +  o(l), 

and  so  Jn(n)  =  2~ ^  S~  '"^ ir-^  T (^) n~  ■-  +  o  (n~ ^). 

To   obtain   the  second   approximation  to  Jnin),  we  obser^ 
that,  when  6  is  small, 

(6«^)^sin^         (i-i&^+^e^-...){i-^^e-^+^L^e^-...f 

Consequently,  if  </) "  ^'  {(1-^0^6'^  ~  efj  ~  "■^'  ^'^^' 

we  have  /o  (0)  =  6  ~  ^  ^  35,  Also,  as  in  the  case  of/i  (</>),  we  assume§ 
for  the  moment  that  /2(^)  has  limited  total  fluctuation  in  the 
range  (0,  tt).  The  application  of  Bromwich's  theorem  is  therefore 
permissible,  and  we  deduce  that 

Lim  ni  ["(^  -  -^  sin  (?i</)) ./,  {j>)d<l>  =  3^  2  "  ^-  T  (|)/35, 

M-s-oo  J  0 

*  A  formal  proof  will  be  given  in  §5a  that /j  (^)  is,  in  fact,  monotonic  and 
decreasing  (we  use  the  term  decreasing  to  mean  non-increasing). 

t  Euler's  result  that    /     ip''^^~'^  sin  \p  cl^p —  V  [m]  sin  {\mir) ,  when   -\<m<l,  is 


well  known. 

J  Bromwich,I?i/ini<e  Series,  p.  444,  proves  that,  if  f{<p)  has  limited  total Jiuetua- 

f  ^  sin  Hd) 
tion  in  the  range  (0,  h),  where  6>0,  and  if  U,^—  I     — - — f((p)d(p,  then 


H-^ao  «-*oo  J  0         W  J  0  f 

but  his  analysis  is  equally  applicable  to  the  more  general  integral 
V^=n'"'  i    (p'^-'^  sin  {7i4>) .  f  (^)  d(p        (-l<m<l), 


and  hence 

Lim  F„=Lim    I      i/^^-i  sin  i// ./(i///?i)(7i/'=/(0)  I     f-^  sin  xj^  df. 
rt^-x  M-*-Qo  Jo  J  i) 

%  A  formal  proof  will  be  given  in  §  5  b  that/^  (<p)  is  monotonic  and  increasing. 


that  is 


of  equal  order  and  argument 
["(/)-«  sin  (7i(/)) .  /;  (<^)  ^0  =  3*  2  -  ^  ?i  -  *  r  (|)/35  +  o  {n  " '), 


45 


and  so 


^  8     (  f'^  sini/r  r'-^  sin^/r        ] 


(6f) 

-r 

??7r  j  0 


</>  ~  I/2  (</))  sin  (7?</)) .  f/<^  +  0  (/i-2) 


secon 
r  such 

sin-^lr 


Now,  by  the  second  niean-vahie  theorem,  there  exists  a  number  a 
exceeding  nir  such  that 


liTT         "^^ 


dyjr 


1 

(nirf 


sin^p•  d/\jr  I  <  2(mr)    •', 


and  so  we  have  at  once  that,  when  n  is  large  and  real, 
^»  (n)  =  ^T—^ ^  +  0  (n  -  -^ ), 

which  is  the  result  to  be  established.  T(3  obtain  a  closer  approxi- 
mation by  these  methods  would  necessitate  some  very  tedious 
integrations  by  parts. 

3.     We  next  consider  the  approximate  formula  for  Jn  (n).     It 
is  immediately  deduced  from  the  Bessel-Schlafli  integral  that 


j:/(7i)=-        sin  (9  .  sin  71  (^  -  sin  (9) .  c?^ 


TT        Jo 

Now  we  get,  on  integrating  by  parts, 

rsinhde-^^^  +  ^'^'^'^^dd 

Jo 


~     nJo    1 


sinh  6       d 


^-n(d  +  sinhd)^^0 


1 

nj  0 


+  cosh  6 '  dd 


^r  e-''Ud  =  0{n-% 
2njQ 


46  Mr   Watsoji,  Bessel  functions 

Hence  /„'  (n)  =  -  I    , -p^  sin  ii6  cl(b  +  0  («~-), 

TT  /  0  1  —  cos  0  ^    ^ 

where  ^,  as  previously,  stands  for  6  —  sin  6. 

Now,  if  fs  ((}>)  =  ^^  sin  ^ .  (1  -  cos  e)-\  then/:  (0)  =  2^  3  '  ^  and 
/^{(fi)  has  limited  total  fluctuation*  in  the  range  (0,  tt). 
Hence,  applying  Bromwich's  theorem  we  have 

o  f'^  sin  n(6  .  , ,,  , ,       ,  ,.,  f'"  sin  ilr  -  ,  ,,, 

J  0  (p''  Jo         T^-- 

and  so  J",/  (n)  =   „   ^^^  +  o  (?i "  *)  +  0  (n-'),  ^ 

TTIV- 

when  n  is  large  and  real ;  and  this  is  equivalent  to  the  result 
stated  in  §  1.  The  approximation  could  be  carried  one  stage 
further  (as  in  §  2),  but  it  seems  hardly  necessary  to  give  the 
analysis. 

4.  As  an  example  of  the  necessity  for  the  caution  which  has 
to  be  taken  in  approximating  to  integrals  with  rapidly  oscillating 
integrands,  it  may  be  remarked  that  some  of  the  earlier  writers 
mentioned  in  §  1  assumed  that  when  x  and  n  are  large  and  nearly 
equal  [in  fact,  when  \x  —  n\  =  o (n^)],  then  Airy's  integral 

An  («?)  =  -[    cos  [nd  -  cc  {6  -  ^6')}  cW 

is  an  approximation  to  Bessel's  integral  for  J^  (^).  This  assumption 
is  correct,  and  it  happens  that  the  first  tivo  terms  in  the  asym- 
ptotic expansions  of  An  (;»)  and  Jn  {oc)  are  the  same. 

But  Airy's  integral  for  An  {not)  is  not  an  approximation!  to 
Jn  (no)  when  a  is  fixed  and  0  <  a  <  1,  while  n  — >  x  . 

To  establish  this  statement  we  use  Carlini's  formula:}: 

Jn  (na)  ~  -_ 

{1  +  ^/(l  -  a2)}» .  (1  _  a')i  V(27rw) 

(valid  when  0  <  a  <  1),  and  after  observing  that  we  ma^-  write 

An  (yia)  =  -  I  —  )    /    cos  IW  (mw  +  lu^)]  dw, 
7T\naJ  J Q         '■^     ^  n       ' 

*  A  formal  proof  will  be  given  in  §  5  c  that  /I,  [cp)  is  monotonic  and  decreasing. 

t  For  example,  the  arguments  given  in  the  P/u7.  Mag.,  August  1908,  p.  274, 
to  justify  the  approximation  seem  to  me  to  be  as  applicable  to  the  second  case  as 
to  the  first. 

X  A  translation  of  Carlini's  memoir  (published  at  Milan,  1817)  was  given  bv 
Jaeobi,  Astr.  Nach.  xxx.  (1850);  Qes.  IVerl^e,  vii.  pp.  189—245.  See  p.  240  for 
the  formula  quoted. 


of  equal  order  and  argument  47 


2?t(l-a) /37r\^  ( nOL\\:  ^ 


where  m  — (  —  J   ,  w 

IT         \naj 

X 

we  use  Stokes'  asymptotic  formula* 

cos  IItt  (mw  +  10^)]  dw  ~  2  "  ^  (3m) "  *  exp  |  -  ir  (|w)^}, 

.'0  " 

valid  for  large  values  of  m. 
This  process  gives 

exp  { -  in  2^  a  ~  ^  (1  -  g)^| 

{2a(l-a)}V(27rn) 


^"  ("«)'-  r^      .,  Mi    ./ 


Hence  ./nOi«)  ^ /JgLVe^xM 

where 

X  («)=  V(l  -  «•-■)  +  log  «  -  log  {1  +  V(l  -  «"^)l  +i«"'(2  -  2a)i 
Since  %  (i)  =  -02047, 

a  rough  approximation  to  .7io„o  (500)/J.iooo  (500)  is  (f  )*  e-"'*l 

5.  We  now  prove  the  monotonic  properties  (valid  for  0  $  ^  ^  tt) 
stated  in  §§  2,  3  : 

(A)  To  prove  that  /^  (</>)  =  i  (6<^)*  sin  ^  .  (1 -cos  ^)--  is  a 
decreasing  function,  we  have 

d  ...  .,.,..,  _  (3  +  2cos^)<^^^,(^)' 

re  ^^^'  ^^^/^  ^-  -  — (i-cos^)3 — ' 

where     g,  (6)  =  [5  sin  0(1-  cos  ^)/(9  +  G  cos  9)]- 6  +  sin  6, 

so  that 

(y/  ((9)  =  -  6  (1  -  cos  ey/{9  +  6  cos  6)'  ^  0,   and  ^r  (0)  =  0. 

We  now  see  that  gi{0)^0,  and  so  f/{<ji)^0,  which  is  the 
result  stated. 

(B)  To  prove  that 

/,{<!>)  =^-^ [(8/6*)  -  {(^^ sin  6/(1  -  cos  Of]] 

is  an  increasing  function,  we  first  prove  two  subsidiary  theorems, 
namely  : 

B  (i).     If  c  =  cos  0,  s^  sin  0,  then  the  function 
g,  (0)  =  (85  +  163c  +  84c-  +  18c^)  (/>  -  is  (1  -  c)  (149  +  157c  +  44c0 
is  not  positive. 

*  Math.  Papers,  ii.  p.  343.  The  result  may  also  easily  be  derived  from 
NicholBon's  expression  of  Airy's  integral  in  terms  of  Bessel  functions  of  order  ±  J, 
Phil.  Mag.,  July  1909,  pp.  6—17, 


48  ilf?'  Watson,  Bessel  functions  etc. 

B  (ii).     The  function 

g,{d)  =  2s (7  +  3c) (/)-^  -  3 (3  +  2c) (1  _  c)^  <^  +  f s  (1  -  c? 
is  not  positive. 

To  prove  B  (i)  we  observe  that 

^  [g,  (6)1(85  +  163c  +  84c-^  +  ISc^} 

=  -  s-  (1  -  cy  (644  -I-  416c  +  60c-)/(85  +  163c  +  84c-  +  18c--)- 

The  denominator  may  be  written  in  the  form 
I873  +  3O72  +  497  -  12 
where  7  =  1  +  c,  and  so  the  denominator  changes  sign  once  on 


i 


when  0  <  ^  <  TT,  say  a,t  6  =  ^.     Hence 

g,  (^)/(85  +  163c  +  84c^  +  18c0 

decreases  from  0  to  —  czd  and  then  from  +  00  to  tt  as  ^  increases 
from  0  to  yS  and  then  from  /3  to  tt.  .Hence  gi(6)  cannot  be 
positive. 

To  prove  B  (ii)  we  observe  that 

^  [15^5  (0)1  [s  (7  +  3c)|]  =  (1  -  c) g,  (0)/{2  (1  +  c)  (7  +  dcf}  ^  0, 

by  B  (i) ;  and  so  g^  (6)  ^  g^  (0)  =  0,  as  was  to  be  proved. 
To  prove  the  main  theorem,  we  have 

where  g  (6)  =  {(3  +  2c)  f  ^  -  s  (1  -  c)  (f)^  (1  -  c)-\ 

Now  g'  (6)  =  -  ^a  (0)  f  "  (1  -  c)-  ^  0, 

so  that  g(0)^g (0)  =  64/6i 

and  so  //  (^)  ^  0,  as  was  to  be  proved. 

(C)     To    prove   that   f(<j>)  =  (f>i  sin  6  .  (1  -cos  (9)-i    is  a  de- 
creasing function,  we  have 

^^  =  i<^-ni-cos^)-^3(^), 
where  g^  (0)  =  sin  ^  (1  -  cos  e)-S(d-  sin  6). 

Since  g/  ((9)  =  -  2  (1  -  cos  ey  we  may  use  the  arguments  of  (A) 
to  prove  the  truth  of  theorem  (C). 


Mr   Watson,  The  limits  of  applicability  etc.  49 


The  limits  of  applicability  of  the  Principle  of  Stationary 
Phase.     By  G.  N.  Watson,  M.A.,  Trinity  College. 

[Received  22  November  1916.] 

1.     The  method  of  approximating  to  the  value  of  the  integral 

».  =  ^r-        cos  [m  {oD  —  tf(m)}]  dm, 

Ztt  .'  0 

where  x  and  t  are  large,  by  considering  the  contribution  to  the 
integral  of  the  range  of  values  of  m  in  the  immediate  vicinity 
of  the  stationary  values  of  m  {.r  —  tf(m)],  is  due  to  Kelvin*,  though 
the  germ  of  the  idea  may  be  traced  in  a  paper  published  nearly 
forty  years  earlier  by  Stokes  f. 

Kelvin's  result  is  that,  if  m{x  —  tf(m)]  has  a  minimum  when 
m  —  [x>  0,  then,  as  ^  — >  oo  , 

u  ~  i^-nt)  -  *  {-  ,xf"  (/.)  -  2/-'  (fx)]  -  *  cos  [t,x\f'  (/x)  +  iTr} ; 

and  this  result  has  imjDortant  applications  in  connexion  with 
various  problems  of  mathematical  physics  J. 

Kelvin,  in  his  analysis  of  this  interesting  asymptotic  formula, 
takes  for  granted,  on  physical  groimds,  the  validity  of  a  certain 
passage  to  the  limit.  This  process  requires  justification  from  the 
purely  mathematical  point  of  view  ;  and  the  necessary  justification 
is  afforded  by  a  convergence  theorem  due  to  Bromwich§.  This 
theorem  plays  the  same  part  in  dealing  with  integrals  as  an 
analogous  theorem,  due  to  Tannery |j,  plays  in  connexion  with 
series. 

The  special  form  of  Bromwich's  theorem,  which  is  required  in 
the  rigorous  investigation  of  Kelvin's  theorem,  may  be  enunciated 
as  follows : 

If  f{x)  be  a  function  of  x  with  limited  total  fluctuation  in  the 
range  x  ^  0,  and  if  7  be  a  function  of  n  such  that  ny  —^  00  as 
n  —^  00  ,  then,  if  —  1  <m<l, 

*  Phil.  Mag.,  March  1887,  pp.  252—255  {Math,  and  Physical  Papers,  iv. 
pp.  303—306). 

t  Camh.  Phil.  Trans,  ix.  (1851),  p.  175  (Math,  and  Physical  Papers,  11.  p.  341). 

t  See  Macdonald,  Phil.  Trans.  210  a.  (1910),  pp.  134—145. 

§  Bromwich,  Theory  of  Infinite  Series,  p.  444.  In  the  special  case  7H  =  0,  which 
is  explicitly  considered  by  Bromwich,  the  result  is  important  in  the  investigation 
of  Fourier  series  by  the  method  of  Dirichlet.  The  theorem  given  by  Bromwich  on 
p.  443  is  equally  applicable  to  the  more  general  case. 

II  Fonctions  d'une  variable,  p.  183. 

VOL.   XIX.    PT.   I,  4 


50  3Ir   Watson,  The  limits  of  applicahility 


i 


-co 

x'"^-^f{x)  sin  nxdx  ->/(+  0)       t"'-'  sin  tdt 

.'  0 

=/(+ O)r(w)  sin  I  m-TT. 


[//  0<m<  1,  the  sines  may  be  replaced  throughout  by  cosines ; 
and,  if  ny—^  a  as  n-^  oo  ,  where  a  is  finite,  the  infinity  in  the  upper 
limit  of  the  integral  must  be  replaced  by  o-.] 

As  the  formal  analytical  proof  of  a  theorem*  slightly  more 
general  than  Kelvin's  theorem  is  quite  simple,  and  as  sufficient 
general  restrictions  to  be  satisfied  by  the  function' /(?n)  are 
apparent  in  the  course  of  the  investigation,  it  seems  to  be  worth 
while  to  place  the  theorem  on  record.  It  is  applicable  to  all 
kinds  of  stationary  points,  whereas  Kelvin  considered  only  cases 
of  true  maxima  or  minima  of  the  simplest  type. 

2.  The  main  theorem  which  will  be  proved  in  this  paper  is 
as  follows  f: 

Let  a,  /3  be  any  numbers  {infinity  not  excluded),  possibly  depending 
on  the  variable  n,  such  that  the  real  function  bt  —  tf(t)  has  only  one 
stationary  value  in  the  range  a^t^ ^,  at  t  =  /m,  b  being  independent 
of  n.  Let  the  first  r  differential  coefficients  with  regard  to  t  o/i| 
bt  —  tf(t),  be  continuousX  in  a  range  of  values  of  t  of  which  t  =  fxis^^ 
an  interior  point,  it  being  supposed  that  the  last  of  them  is  the  lowest 
which  does  not  vanish  at  t=  /j,,  so  that  r  ^  2. 

Let  F  (t)  be  a  real  function,  continuous  when  a<t  <  /3,  except 
possibly  at  t  =  fi,  and  let 

Urn  F{t).{t-  ixY  =  A,    Lim   F (t) . {tju  - 1)"^ :=  A„ 
where  A,  Ay^  are  not  zero ;  for  brevity,  let  (1  —  A,)/?-  =  m.. 
Then,  if  the  function 

F(t).\bt-tf{t)-,ji'^f{,,)Y-^-.\b-tf'{t)-f{t)\-^ 
has  limited  total  fiuctuation^  in  the  range  a^i^/9,  and  if 

I  nb^  -  n^f(,8)  -  n/ii'f  (/.)  | ,     |  nba  -  noif  (a)  -  ntif  (^)  j 
both  tend  to  infinity  luith  n,  the  approximate  value  of  the  integral 

/  s  ^  [  F{t)  cos  [bnt  -  ntf(t)}  dt, 

*  For  the  connexion  between  this  theorem  and  a  problem,  due  to  Riemann 
(Werke,  p.  260),  which  has  been  discussed  by  Fej^r  {Comptes  Rendus,  November 
30,  1908,  and  a  memoir  published  by  the  Academy  of  Budapest  in  1909)  and  by 
Hardy  (Quarterly  Journal,  xliv.  1913,  pp.  1—40  and  242 — 263),  see  §4  below. 

t  It  is  convenient  to  modify  Kelvin's  notation. 

X  It  is  necessary  iox  f{t)  to  have  a  continuous  first  differential  coefficient  when 

§  If  the  fluctuation  depends  on  n,  it  must  be  a  bounded  function  of  n  as  7i-».x  . 


of  the  Principle  of  Stationary  Phase  51 

wlien  n  is  large,  is 
r- 1)  !(r !)'"-!  r(m)[^  cos  {nfx,^f'(fi)+  ^em-n-]  +  A,  cos  {nfj:'f'{fM)  + 1 7?m7r}] 

provided  that  0  <  1  —  X  <  r ;  wAere  e  =  ±  1  according  as  bt  —  tf{t) 

increasinq    ,        .         ,  ,  ^  ,.  , 

%s  an  ,  .       f  unction  when  t  >  a,  ana  ??  =  +  1  accoratnq  as  the 

decreasing  n  •      - 

same  function  is    .  .       when  t  <  a.     When  n—^cc    hti  only 

increasing 

such  values  that  cos  [nfM-f  {/u.)}  is  always  zero,  A,  may  lie  in  the 

extended  range  —r<l  —  X<  r.     And,  finally,  F{t)  and  bt  —  tf(t) 

inay  be  infinite  at  t=  a,  ^,  provided  only  that  the  integral  converges 

for  all  sufficiently  large  values  of  n. 

3.     For  brevity,  write  tf  (t)  ^  (f)  (t).     Then  ^  is  given  by  the 
equation 

b-<f>'(,M)  =  0, 

so  that,  when    t  —  fi,  is  sufficiently  small, 

bt  -  tf(t)  -  t(}>'  (/jl)  -(f){t) 

=  {/.f  (/.)  -  </)  (/.)}  -(t-  fxyr'  {t')/r  !, 

where,  by  Taylor's  theorem,  t'  lies  between  fx,  and  t. 
Now  define  a  new  variable  yjr  by  the  equation 

bt-tf{t)  =  fi<l>'{ix)-c},ifi,)  +  f, 

and  let  7,  F  be  the  values  of  yfr  corresponding  to  t  =  a,  t—  /3. 
Noticing  that  /x^'  (/j,)  —  (f>  (fi)  =  fi^f  (/n).  we  have 

J.     cos  {n/M'f  (fi)}  r^„,,,  ,   u 

sin{n/jb^f'  (a)}  T^et/^x    •        ,   7^ 
Zir  J  a 

i/r  being  a  monotonic  function  of  t  when  a  ^t  ^  /x  and  also  when 
fi^t^^. 

Now  e,  7;  have  been  so  chosen  that  e-yjr  and  rj^jr  are  positive 
when  t  >  /M  and  ^  <  //-  respectively ;  hence,  when  ^  —>//,  +  0,  we  have 

df 

ef  '^  {t-  fiY   </)<'■'  {fi)  1  ^  r  !. 


52  Mr  Watson^  The  limits  of  applicahility 

It  follows  that 


^^^^^A'^^'"'""'"'"'-^^^' 


as  i  — >  /Lt  +  0,  where 


^^     -(r-l)!(r!)^ 


</><'•)(;.)  11  (/,<'•)  (/.)!}--' 


Since  j  wF  ]  — >  oo  with  h,  by  hypothesis,  we  deduce  from  Bromwich's 
theorem  that 


cos 


r  p(-)o  (If  reoo 

0  ^<*>  sin  «+  St  '^  ~  ""  '"'  .1 .    <^'>"'"'  -n  ^'^- 
Writing  %e  s  &>,  we  get 

re  CO  rcc 

I       (%^)™~^  COS  %<^%  =  e  I     &>'"~^  cos  ctx^o)  =  eF  (??i)  cos  -| 

and  similarly 

(%e)'"~^  sin  %c^%  =  F  (m)  sin  ^7«7r. 


7?2-7r, 


In  like  manner,  when  t-^  /jl-  0, 
dt 


and  so,  since  1 717  |  — >  oo  with  n,  we  have 

/,  ^  (*)  sm  "^  4  ^'^  ~  (-)•■  -'" ^■^i  „    (^'''•""'  sin  ^<'^- 

Collecting  our  results,  we  see  that  the  first  approximation  to 
/  is 

/  ~  [{AKe  +  (— )'■  AiKr]}  cos  ^iutt  cos  {nfM-f  (fM)] 
-  [AK  +  (-)'•  A^K]  sin  |m7r  sin  {^i/^y  (/i)}] 
=  A  [cos  (w^-/'  (/u.)  +  ^emir]  +  J.i  cos  {ufx^f  (fi)  +  ■|?;??i7r}] 

(r  -  1)  !  (r  !)'»-!  F  (m) 
^     27rw'^{|<^<'-'(;u)j}'"    ' 

and  this  is  the  result  stated. 

The  formula  fails  to  be  effective  in  the  neighbourhood  of  those 
values  of  n  for  which  the  expression  in  [  ]  vanishes,  as  the  error 
in  the  approximation  then  becomes  comparable  with  the  approxi- 
mation obtained. 

[It  is  evident  that  if  the  cosine  in  the  integral  defining  /  may 
be  replaced  by  a  sine,  then  the  cosines  in  the  approximation  are 
replaced  by  sines.] 


of  the  Principle  of  Stationary  Phase  53 

Cases  of  practical  importance  are  those  in  which  A  =  Ai  and 
t  =  fj>  is  a,  true  minimum  or  maximum  of  bt  —  tf{t),  so  that  e  and  r) 
are  both  +  1  or  both  —  1.     The  formula  then  is 

A  cos  {n/M'f'ifM)  ±  ^mTr} .  (r  -  1) !  (r  !)'»-i  T(m) 
~  7r?i"*  { I  (^e-)  (fi)  I  1*'^  • 

If  nV  or  ny  tend  to  finite  limits,  the  gamma  functions  have  to 
be  replaced  by  incomplete  gamma  functions ;  and  if  one  or  other 
tends  to  zero,  we  modify  the  approximation  by  writing  zero  for 
A  or  A^  respectively  in  the  general  formula. 

The  general  result  reduces  to  Kelvin's  formula  when  r  =  2, 
X  =  0,  711=  h,  and  e  =  ?;  =  1,  provided  that  (with  Kelvin's  notation) 
x/t  is  constant.  In  that  case,  a  sufficient  condition  for  the  validity 
of  the  formula  is  that 

^  [{{ma^/t)  -  mf{m)  -  i^-f  (;.)}*]-^ 

should  have  limited  total  fluctuation  when  m  ^  0. 

If  X  were  a  function  of  t,  Bromwich's  general  theorem  {loc.  cit., 
p.  443)  would  have  to  be  used,  and  the  enunciation  of  sufficient 
conditions  (even  in  their  simplest  form)  for  the  validity  of  the 
formula,  would  be  exceedingly  laborious.  The  reason  for  this  is 
that  (with  the  notation  employed  in  this  paper)  -^  and  F  (t)  dt/dyfr 
would  both  be  functions  of  n. 

4.  The  problem  of  Riemann  (see  §  1  above)  essentially  consists 
in  obtaining  an  approximation  for  integrals  of  the  type 


/■""     /  ,x  cos  sin  nt. 


when  n  is  large  and  a-'  (t)—>oo  as  t—>  0. 

These  integrals  are  expressible  by  integrals  of  the  type 


t-'pit)^"^^  {nt+  (7(t)]dt, 
Jo  sm  !  ' 


so  that  the  problem  is,  at  first  sight,  very  similar  to  that  discussed 
in  %  2—3. 

There  is  however  an  essential  difference,  namely  that,  in  the 
problem  we  have  discussed,  ntf{t)  owes  its  large  rate  of  increase 
(which  balances  the  rate  of  increase  of  nbt  at  the  stationary  point) 
to  the  large  factor  n,  whereas,  in  the  problem  attacked  by  Fejer 
and  Hardy,  the  function  a  (t)  owes  its  large  rate  of  increase  to  the 
infinity  of  a  (t)  at  ^  =  0.  In  our  problem  fj,  is  fixed,  whereas  in 
the  other  problem  the  stationary  point  of  nt  —  (T{t)  tends  to  zero 
as  ?i  — *  00  .  It  seems  to  be  this  difterence  which  accounts  for  the 
somewhat  elaborate    investigation  given   by    Hardy   and   which 


54  Mr   Watson,  The  limits  of  applicahility 

makes  the  theorems  of  Fejer  and  Hardy  rather  deeper  than  the 
theorem  of  §§  2 — 3. 

It  should  be  pointed  out  that  there  is  one  integral  which  can 
be  regarded  as  coming  under  either  head,  namely*, 

/•oo 

/         X  sm  . 

I    X  "       (nx  +  ax~n  dx, 

j  0         cos  ^  ^      ' 

where  n  is  large,  a,  \,  and  r  are  positive  and  X  and  r  are  chosen  so 
that  the  integral  converges.  [For  the  sine-integi-al,  the  conditions 
for  convergence  are  0  <  X  <  r  +  1.]  As  the  integral  stands  it  is 
of  the  type  discussed  by  Fejdr  and  Hardy,  with  a  variable 
stationary  point  where  x''+^  =  arjn.  But  if  we  make  the  sub- 
stitution 

and  then  write  v  for  n''''<»'+i',  it  becomes 


j,(A-i)/r[   t-^^^^'^lvit  +  at-'y^dt, 

Jo        cos  ^    ^  ^•' 


which  is  of  the  type  discussed  in  this  paper,  having  a  fixed 
stationary  point  where  t  =  {ray/^''+^K  The  reader  will  have  no 
difficulty  in  deducing  the  approximate  formula  by  either  method. 

5.     As   an   example   of  the  apparent   inapplicability   of  the 
methods  of  this  paper  consider  the  integral  of  Bessel  for  Jn(x) 

when  n  and  x  are  both  large  and  a;  —  ?i,  is  0  (n^). 
The  integral  is 

1  f'^ 
Jn  (x)  =-  I    COS (n6  —  X sin  0) dd, 

and  the  stationary  point  is  given  by   cos  0  =  nlx;   let   the   root 

of  this  equation  be  0  =  /j,,  and  let  x  =  n  +  an^  where  a  >  0 ;  when 
n  is  large  we  have 

In  considering  |    cos  (n0  -  x  sin  0)  d0,  we  write 

X  =  n0  —  xsin0  —  (n/j,  —  x  sin  /i,), 
and  the  last  integral  is  expressible  by  integrals  of  the  type 
»(tan^-M)  cos      d0   , 
sin  ^  dx 


f 

Jo 


*  I  am  indebted  to  Mr  Hardy  for  suggesting  that  the  integral  in  which  a-  (t)  =  llt 
can  be  reduced  to  an  integral  of  Kelvin's  type. 


of  the  Principle  of  Stationary  Phase  55 

Now  tq—'^~  ^  co^  6  o^x{6  —  ix)  sin  ix, 

when  6  f^  fM  and  -^r^^oo^iw  fx  .{6  —  ^y. 
Hence,  as  ;)^;  — >  0, 

Lde  -1 

and 


/, 


0  sin  '^  c^x 

\  /-n'tauM-M)  (•  .  1  rf^")        _1C0S 


/•n'tanM-M)(  .  i  rft'l        _1C0S        , 

Now,  as  ?i  ->  CO  ,  ?i  (tan  /x  —  ^)  ->  i  (2a)^  and  so  the  limiting 
range  of  integration  is  of  finite  length, 

1  d9 
Moreover,  \/{2x  sin  fi) .  ^^  -^ >  —  l  as  ?i  — >  go  ^vhen  %  is  ^■ero, 

1  df) 
that  is,  when  d  ^  fi.    But,  when  6  ->  0,  the  limit  of  \/{2x  sin  A*)  •  %-  ^ 

is 

—  {2  sin  fi  (sin  /jl  —  /xcos  At)|"^/(1  —  cos  /a), 

a^icZ,  a.9  n—>cc,  the  limit  of  this  is  not  —  1  hit  —  2  \/(^) ;  and  so 
we  cannot  infer  that 

•«(tunM-M)f  .  if^6l)       _icos        ,  f'    _icos        , 

|V(2.-sm;.).X-^^~j-%    -^i^^X^^X-j^^X    \sin^^^' 

where  b  is  Lim  •??  (tan  yu,  — /x). 

The  evaluation  of  the  approximate  formula  for  Jn{^)  in  the 
circumstances  under  consideration  consequently  seems  to  require 
more  elaborate  analysis  than  is  afforded  by  the  methods  contained 
in  this  paper. 


56     Mr  Borradaile,  On  the  Functions  of  the  Mouth-Parts  etc. 

On  the  Functions  of  the  Mouth-Parts  of  the  Common  Prawn. 
By  L.  A.  Borradaile,  M.A.,  Selwyn  College. 

[Read  30  October  1916.] 

The  food  is  seized  by  either  pair  of  chelipeds,  or  by  the 
third  maxillipeds,  and  is  usually  placed  by  them  within  the 
grasp  of  the  second  maxillipeds,  though  sometimes  it  is  passed 
directly  to  deeper-lying  structures.  The  second  maxillipeds  are 
the  most  important  of  the  food-grasping  organs.  They  have  three 
principal  movements;  in  one,  the  broad  flaps  in  which  they  end 
open  downwards  like  a  pair  of  doors,  and  with  their  stout  fringes 
gather  up  the  food ;  in  another,  they  rotate  in  the  horizontal  plane 
to  and  from  the  middle  line  of  the  body,  and  thus  narrow  or  widen 
the  gap  through  which  the  food  passes;  in  the  third,  the  bent  distal 
part  of  the  limb  tends  to  straighten,  so  as  to  brush  forward  any 
object  which  lies  between  them.  Frequently  these  movements  are 
combined.  Owing  to  the  facts  that  the  second  maxillij)eds  cover 
the  mouth-parts  anterior  to  them,  and  that  if  they  be  removed 
feeding  is  not  properly  performed  and  usually  not  attempted,  it  is 
difficult  to  trace  the  food  beyond  them,  but  the  following  seems  to 
be  its  fate.  If  it  be  small  in  bulk,  or  finely  divided,  or  very  soft, 
it  is  passed  to  the  maxillules,  by  whose  strong,  fringed  laciniae  it  is 
swept  forwards,  and  probably  caused  to  enter  through  the  slit 
between  the  paragnatha,  into  the  chamber  which  is  guarded  by 
the  upper  and  lower  lips.  If  it  be  tough  or  in  large  masses,  the 
second  maxillipeds  and  maxillules  brush  it  forwards  towards  the 
incisor  processes  of  the  mandibles.  The  action  of  the  latter  is,  by 
rotating  in  a  vertical  plane,  to  tuck  the  food  into  the  gap  between 
the  paragnatha  and  the  labrum.  If  the  mass  be  large,  pieces  are 
torn  off  it  by  this  action.  Finally,  to  enter  the  gullet,  the  food 
must  pass  between  the  molar  processes  and  be  pounded  by  them. 

The  mandibular  palps,  maxillae,  and  first  maxillipeds  appear 
to  play  parts  of  little  importance  in  regard  to  the  food.  The 
palps  are  present  and  absent  in  closely  related  genera,  and  appear 
to  be  disappearing  in  the  higher  Carides.  The  same  is  true  of  the 
lobes  of  the  maxillae,  which  are  in  constant  regular  motion  to  and 
from  the  middle  line,  and  probably  serve  to  restrain  the  action  of 
the  scaphognathite.  The  large  laciniae  of  the  first  maxilliped 
may  have  as  their  function  the  covering  of  the  maxillae  and 
protecting  them  from  the  food.  The  labrum  undergoes  active 
movements,  whose  function  is  probably  to  aid  in  keeping  the 
food  under  the  action  of  the  mandibles.  The  exopodites  of  the 
maxillipeds  set  up  a  strong  current  forwards  from  the  mouth. 
No  doubt  this  aids  in  carrying  away  the  exhausted  water  from 
the  gill  chamber  and  the  excreta  from  the  tubercles  of  the  green 
glands.  Into  the  same  current  particles  which  have  been  taken 
as  food  are  from  time  to  time  rejected  by  the  forward  kicking 
of  the  second  maxillipeds. 


CONTENTS. 


PAGE 

A  self-recording  electrometer  for  Atmospheric  Electricity.  By  W.  A. 
Douglas  Ritdgb,  M.A.,  St  John's  College     .         .        . 

On  the  expression  of  a  number  in  the  form  ax^-\-hy^-^cz--\-dv?.  By 
S.  Ramanujan,  B.A.,  Trinity  College.  (Communicated  by  Mr 
G.  H.  Hardy)  .        .        .        .        .        .        .        .        .        •       H   j 

An  Axiom  in  Symbolic  Logic.  By  C.  E.  Van  Hokn,  M.A.  (Com- 
municated by  Mr  G.  H.  Hardy) 22 

A  Reduction  in  the  number  of  the  Primitive  Propositions  of  Logic.  By 
J.  G.  P.  NicoD,  Trinity  College.  (Communicated  by  Mr  G.  H. 
Hardy)     . 32   " 

Bessel  fuiictions  of  eqiial  order  and  argument.     By  G.  N.  Watson,  M.A., 

Trinity  College  .         . .       42  ' 

The  limits  of  applicability  of  the  Principle  of  Stationary  Phase.     By 

G.  N.  Watson,  M.A.,  Trinity  College   .         .         .        .         .         .       49 

On  the  Fionctions  of  the  Mouth-Parts  of  the  Common  Prawn.     By  L.  A. 

BoRRADAiLE,  M.A.,  Selwyn  College .         .         .         .         .         .         .56 


PROCEEDINGS 


OF   THE 


CAMBRIDGE   PHILOSOPHICAL 
SOCIETY 


VOL.   XIX.     PARTS   IL,  III. 

[Lent  and  Easter  Terms  1917.] 


AT    THE    UNIVERSITY    PRESS 
AND   SOLD  BY 
DEIGHTON,  BELL  &  CO.,    LIMITED, 
AND  BOWES  &  BOWES,  CAMBRIDGE. 

CAMBRIDGE   UNIVERSITY   PRESS, 
C.    F.    CLAY,    MANAGER,   FETTER  LANE,   LONDON,   E.C.  4 

1917 
Frice  Two  Shillings  and  Sixpence  Net 

October  1917. 


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addressed  to  one  of  the  Secretaries, 

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PROCEEDINGS 

OF   THE 


The  Direct  Solution  of  the  Quadratic  and  Cubic  Binomial 
Congruences  with  Prime  Moduli.  By  H.  C.  PocKLiNGTON,  M.A., 
St  John's  College. 

[Received  22  January  1917:  read  5  February  1917.] 

1.  The  solution  of  congruences  by  exclusion  methods, 
although  easy  enough  when  the  modulus  is  moderately  large, 
becomes  impracticable  for  large  moduli  because  the  labour  varies 
as  the  modulus  or  its  square  root.  In  a  direct  method  the  labour 
varies  roughly  as  the  cube  of  the  number  of  digits  in  the  modulus, 
and  so  remains  moderate  for  large  moduli.  The  object  of  this 
paper  is  to  develop  the  direct  method.  We  take  or  =  a,  mod.  p, 
first,  discussing  the  cases  where  p  =  4<m  +  3  and  p  =  Sin  +  5  in 
§  2  and  that  where  p  =  8ni  +  1  in  §  3.  We  next  take  ar^  =  a  and 
discuss  the  cases  where  p  =  Sm  +  2,  />  =  9m  +  4  and  p=i9m  +  7 
in  §  4  and  that  where  p  =  dm  +  1  in  §  5. 

2.  Throughout  the  paper  we  suppose  the  modulus  to  be 
p  where  p  is  prime*.  If  p  is  of  the  form  4wi  +  3  the  solution  of 
^•-  =  a  is  X  =  ±  a'^'^\  If  p  is  of  the  form  8m  +  5  the  solution  is 
^  =  ±  a'"+i  provided  that  a-'^+^  =  1.  But  if  not,  a'^"*+i  =  -  1,  and 
as  2  is  a  non-residue  4-'"+'  b  -  1 ;  so  that  (4a)-»*+^  =  1  and  we  have 
2/  =  +  (4a)'"+^  as  the  .solution  of  y-  =  4o.     Hence 

x=±y;2  or  x=±{p+  y)/2 

is  the  solution  of  x-=a.  These  values  of  x  can  be  calculated 
without  serious  difficulty  by  repeated  squaring  (followed  by  division 
by  the  modulus  to  find  the  remainder)  and  multiplication  of  the 
numbers  so  found  (again  followed  by  division). 

*  Hence  if  it  is  composite  we  must  factorize  it  and  solve  the  congrnence  for 
each  of  the  different  prime  factors. 


VOL.  XIX.  PARTS  II.,  III. 


58     Mr  Pocklington,  The  Direct  Solution  of  the  Quadratic 

3.  Put  D=—a,  so  that  we  have  to  solve  cc^+  D  =  0  where  D 
is  positive  or  negative  but  not  divisible  by  p.  Let  t^  and  Mj  be  so 
chosen*  that  t^  —  Du-^-  =  N  is  a  quadratic  non-residue  of  p,  and 
let 

tn  =  [{t,  +  U,  ^Dr  +  (t,  -  U,  sjDY]l% 

These  numbers  are  clearly  integral.     Also 

by  use  of*  which  (at  first  with  m  =  n)  we  can  find  the  remainders 
of  tn  and  Un  to  our  modulus  without  serious  difficulty  even  when 
n  is  large.     We  also  have  tn  —  Dun  =  N'^. 

Supposing  that  p  is  of  the  form  4m  +  1,  we  have  D  a  quadratic 
residue  of  'p,  and  tp  =  t^P  =  ^i,  Up  =  u-pD^P~'^''i-  =  u^ ;  and  now 

ti  =  tp^^t-^  +  Dup_^Ui^,     tij  =  tp_iUi  +  t-^Up_^ 

give  on  solution  tp_^  =  1,  Up_-^  =  0.     Let  p-l  =  2r.     Then 

0  =  l/^_i  =  2trUr 

shows  that  either  t,.  or  u,.  is  divisible  by  p.  If  it  is  Ur  we  put 
r  =  2s  and  proceed  similarly.  We  cannot  have  every  u  divisible 
by  p,  for  u^  is  not.  We  cannot  be  stopped  by  having  u,n  =  0  with 
m  odd,  for  we  always  have  4,,^  -  Dum''  =  N''\  and  this  would  then 
give  ^,„'  congruent  to  a  non-residue.  But  if  m  is  even  we  can 
.proceed  further.  Hence  when  we  are  stopped  we  must  have 
t,n  =  0.  This  gives  -  Bii,,^  =  N''\  and  as  -  D  is  a  residue  m  must 
be  even.  Putting  m  =  2n  we  have  0  =  ^^  =  tn'  +  Bun',  so  that  the 
solution   of  a^  +  D  =  0  is  got  by  solving  the  linear  congruence 

UnX  =z  +  tn- 

In  applying  the  method,  if  n  is  the  largest  odd  number  con- 
tained m  p-l  we  first  work  to  get  the  suffixes  n,  and  then  the 
suffixes  2n,  4<n,  8n,  etc.  Thus  in  the  case  of  cc''  +  2  =  0,  mod.  41, 
we  see  that  ^j  =  3,  Ut,  =  1  is  suitable,  and  we  find  t.  =  11,  u.,  =  6  ; 
t,  =  29,  u,=  9;  t,=  23,  u,=  15;  t^o  =  36,  u,o  =  34  ;'  t.^  =  0.  '  The 
solution  of  34^  =  36  is  a;  =30;  and  so  the  two  solutions  of 
*•-  +  2  =  0  are  .x=±  30,  mod.  41. 

^4.  If  p  is  of  the  form  Sm  +  2  the  only  solution  of  x'  =  a  is 
a;  =  l/a'«.  If  p  is  of  the  form  9m  +  4>  one  solution  is  cc  =  !/«'"■  and 
if  of  the  form  9m  +  7  one  is  .x  =  a^+\  The  other  solutions  are 
got  frojn  this  by  multiplying  by  (-  1  +  6)12  and  (-  1  -  e)/2,  where 
t/-  +  3  =  0,  a  congruence  which  we  have  shown  how  to  solve. 

*  We  have  to  do  this  by  trial,  using  the  Law  of  Quadratic  Eeciprocity,  which 
"  •!  ?f  i'"  *^J®  method.  But  as  for  each  vakie  of  n  half  tire  vahies  of  /  are 
.suitable,  there  should  be  no  ditlficulty  in  finding  one. 


and  Cubic  Binomial  Congruences  with  Prime  Moduli      59 

5.  Let  8  be  the  arithmetical  cube  root  of  a,  which  we 
assume*  not  to  be  a  cube.  Findf  ti,  Ui,  Vi  such  that  the  norm 
N  =  t{^  +  au^^  +  a'-Vi"  —  SatiU^i\  of  the  algebraic  number 

is  a  cubic  non-residue  of  p.  We  see  that  as  a  is  a  cubic  residue 
of  p  we  have  U^'  =  ti-\-  UiS+  i\8'^,  so  that  if 

U^'-^  =  tp-i  +  Up-i  8  +  %_i  8- 

we  have  iip^i  =  Vp_i  =  0.  Now  taking  ?7'"  where  m  is  in  turn 
(p— l)/3,  (p  —  l)/9,  etc.  we  see  that  we  cannot  always  have 
u,n  =  v„i  =  0.  Let  f/""^  be  the  last  of  this  series  for  which  this 
happens.  Then  m  is  divisible  by  3,  for  otherwise  the  norm  of  [7"\ 
which  reduces  to  t^,  would  be  congruent  to  the  non-residue  iY'". 
Putting  m  =  on  we  have 

tsn  =  tn    +  <Kln    +  a-^n    +  QatnUnVn, 
0  =  Usn  =  3  (tn'Un  +  ^^tnVn'  +  «",rWn), 
0  =  V.,n  =  3  {taUn'  +  tuVn  +  aUnVn'')- 

The  last  two  give  tn{av^i"'  —  Vn)=  0;  so  that  if  tn  is  not  divisible 
by  p  we  have  w  =  Un/v,,.  as  one  solution  of  n^  =  a,  for  as  Un  and  v,i 
are  not  both  divisible  by  p  this  shows  that  neither  is.  They  also 
give  Vn  (ait^n  —  tn)  =  0,  and  so  w  =  tnjihi  is  a  solution.  Eliminating 
a  from  the  same  two  congruences  we  see  that  the  ratio  A,  of  the 
two  xs,  satisfies  V  +  X  +  1  =  0,  so  that  they  are  distinct.  The 
third  solution  follows  immediately. 

If  however  tn  is  divisible  by  p  the  two  congruences  show  that 
either  Un  or  Vn  is  divisible  by  p.  We  now  have  rtM,i,^=iA'"  or 
a-Vn  =  -Y".  In  either  case  n  must  be  divisible  by  3  as  before,  and 
we  have  as  one  solution  x  =  N^'jun  or  x  =  aVnjN''  respectively, 
where  r  =  n/S. 

*  Simply  because  of  the  way  in  wliich  for  the  sake  of  shortness  we  are  stating 
the  method. 

t  This  again  must  be  done  by  trial.  In  order  to  use  the  Law  of  Cubic 
Beciprocity  we  must  express  p  in  the  form  ii'  +  xiv  +  v'^,  which  requires  the  solution 
of  ^2  +  3  =  0. 


5—2 


60  M7'  Hardy,  On  a  theorem  of  Mr  G.  Polya 

On  a  theorem  of  Mr  G.  Polya.  By  G.  H.  Hardy,  M.A., 
Trinity  College. 

[Received  and  read  5  February  1917.] 

1.  Mr  G.  P(51ya  has  recently  discovered  a  number  of  very 
beautiful  theorems  concerning  Taylor's  series  with  integral  co- 
efficients and  '  ganzwertige  ganze  Funktionen '.  The  latter 
functions  are  integral  functions  which  assume  integral  values  for 
all  integral  (or  for  all  positive  integral)  values  of  the  independent 
variable.  One  of  the  most  remarkable  of  these  theorems  is  the 
following*:    • 

Suppose  thatg{x)  is  an  integral  function,  and  M{r)  the  maodmum 
of  I  g  {x)  I  for  \x\^r.     Suppose  further  that 

g(0),g{l),g{2),... 

are  integers,  and  that 

lim  2-''^JrM{l')  =  0 (1). 


?'-»-00 


Then  g  {x)  is  a  polynomial. 

Mr  Polya  observes  that,  if  it  were  possible  to  get  rid  of  the 
factor  \/r  from  the  equation  (1),  the  theorem  could  be  enunciated 
in  a  notably  more  pregnant  form,  viz. : 

Among  all  transcendental  integral  functions,  which  assume 
integral  values  for  all  positive  integral  values  of  the  variable,  that 
of  least  increase^  is  the  function  2^. 

Mr  Polya  states,  however,  that  he  has  not  been  able  to  effect 
this  generalisation.  And  my  object  in  writing  this  note  is  to 
show  that  the  generalisation  desired  may  be  obtained  by  a  slight 
modification  of  Mr  P(51ya's  own  argument,  and  without  the 
addition  of  any  essentially  new  idea  to  those  which  he  employs. 

2.  Mr  Polyaij:  reduces  the  proof  of  the  theorem  to  a  proof 
that  the  integral 

T(  \—  ^'    r  g(x)dx 

^'^^~2^ij  x(x-l)(x-2)...{x-n)' 

extended  over  the  circle  \x\  =  r  -  2n,  tends  to  zero  when  ?i  — »  oo  , 

*■  G.  Polya,  '  Uber  ganzwertige  ganze  Funktionen ',  Rendiconti  del  Circolo 
Matematico  di  Palermo,  vol.  40,  1915,  pp.  1—16.  See  also  'Uber  Potenzreihen 
mit  ganzzahligen  Koeffizienten ',  Mathemathche  Annalen,  vol.  77,  1916,  pp.  497— 
513,  where  Mr  Polya  refers  to  a  third  memoir  ('  Arithmetische  Eigenschaften  der 
Eeihenentwicklungen  rationaler  Funktionen',  Journal  flir  Mathematik)  which.  1 
have  not  been  able  to  consult, 

t  Croissance,   Wachstum. 

X  Loc.  cit.,  p.  7. 


Mr  Hardy,  On  a  theorem  of  Mr  0.  Polya 


61 


This  he  proves  by  observing  that  the  modulus  of  J^^  does  not 

exceed 

n\M{r)  ^  V{n  +  \)V{n) 

(r-l)(r-2)...(r-w)  r(2n)  ^  '' 

and  by  an  application  of  Stirling's  Theorem.  In  order  to  com- 
plete the  proof  in  this  manner  it  is  necessary  to  assume  the 
condition  (1). 

If  however  we  suppose  only  that 

lim2-'-ilf(r)  =  0 (2), 

or  J'f(r)  =  o(2'-)    (2'), 

the  proof  may  be  completed  as  follows.     We  have 

where  ^=  2ne'^.     Now 

\x  —  s\  =  V(4?r  —  4?2.s  cos  6 +  8^)'^  2n  —  s  cos  0 
for  1  <s  <.n,  so  that 


de 

.){a)-2)...{a;-n)\\  ' 


U{x-  s) 

1 

if  cos  ^  >  0,  and 


>  n  (2n  -  s  cos  6)  =  (cos  ^f  11  (2«  sec  O-s) 


n  (x  -  s) 
1 


^  n  (2n  -  s  cos  ^)  =  I  cos  ^  i"  n  (27*  |  sec  ^  |  +  s) 
1  1 


if  cos  ^  <  0.     Hence 

where  Kn  =  nl2^''j 


Jn  =  0  {Kn)  +  0  (X„), 

'''  T(2na-n) 
r  {2n(r) 


a'^dO, 


i„  =  „,2»|'V„_£(2^iV,.«rf«, 


and  a  =  sec  0. 

A  straightforward  application  of  Stirling's  Theorem  shows  that 

uniformly  in  6,  where 

^  =  <t>  (0)  =  {2a  -  1)  log  (2o-  -  1)  -  2a-  log  2o-  +  log  o"  +  2  log  2, 
^  =  ^  (^)  =  2(T  log  2o-  -  (2o-  +  1 )  log  (2a  +  1)  +  log  o"  +  2  log  2. 


62 


Mr  Hardy,  On  a  theorem  of  Mr  0.  Polya 


=  2  log  2cr  -  2  log  (2o-  +  1)  +  ^  =  ^  -  2  log  (l  +  ^M  >  0. 


3.  When  6  increases  from  0  towards  -|-7r,  or  decreases  towards 
—  hir,  (T  increases  from  1  towards  oo  .     Also 

^^  =  21og(2c.-l)-21og2^  +  ^=2  1og(l-^^)+-^<0, 

da- 

Thus  <I>  steadily  decreases  and  ^  steadily  increases.     Moreover 

^  (0)  =  0,     ^  (0)  =  4  log  2  -  3  log  3  ; 

and  it  is  easily  verified  that  both  <J»  and  ^  tend  to  the  limit 

log  2  -  1 
when  6  tends  to  ^tt. 

We  thus  obtain,  in  the  first  place, 

i,.=of-'.<--vnj:;y(,^)</«}=o(i). 

Secondly,  we  observe  that,  if  S  is  any  positive  number,  we  have 

^{d)<^{h)  =  -7^<0 

for  h^e^^ir,     -^ir^eK-h. 

Hence  we  may  replace  the  limits  in  Kn  by  -  8  and  h,  the  re- 
mainder of  the  integral  being  of  the  form 

4.  All  that  remains,  then,  is  to  prove  that 
/„  =  ,U2»f  n^^)w(*  =  0(l); 

j_5      1  (2/lcr) 

The  function  ^{6)  may  now  be  expanded  in  powers  of  ^.     We 
find  without  difficulty  that 

where  A  =  log  2  -  ^  >  0. 

It  follows  that 


and  we  have 


h  =  0bn\  ^e-^»^HO(»^*)^^ 
=  0U/n  r  e--^^''^'de\^0{l). 


Mr  Hardy,  On  a  theorem  of  Mr  G.  Polya  63 

The  proof  of  the  theorem  conjectured  by  Mr  Polya  is  thus 
completed. 

5.  Mr  Polya  has  also  proved  an  analogous  theorem  concerning 
integral  functions  which  assume  integral  values  for  all  integral 
values  of  x,  viz.: 

If  ...,g{-2),  g{-l),  g{0\  g{\),  g{2)... 

are  integers,  and 

lim  (^^rVrilf(r)=0 (3), 

then  g  {a;)  is  a  polynomial. 

His  proof  applies,  as  it  stands,  to  odd  functions  only,  its  appli- 
cation to  a  completely  general  function  demanding  the  more 
stringent  condition 


lim 


r-*-co 


^4r^l  'rm(r)  =  0 (3'). 


He  states  that  it  is  possible  to  replace  the  index  |  by  ^  in  all 
cases,  but  that,  as  he  has  not  been  able  to  reduce  the  condition  to 

lim  (^^^y*'ilf(r)  =  0 (3"), 

he  has  not  thought  it  worth  while  to  publish  the  details  of  his 
work. 

A  modification  of  Mr  P<51ya's  argument,  in  every  way  similar 
to  that  which  I  have  made  in  the  proof  of  his  first  theorem, 
enables  us  to  replace  (3)  by  (3")  when  g{x)  is  odd.  The  same 
modification  in  his  unpublished  argument  would,  I  presume,  be 
equally  effective  in  general. 

That  the  number 

3  +  ^/5 


cannot  be  replaced  by  any  larger  number,  and  so  really  is  the 
number  which  ought  to  occur  in  any  theorem  of  this  character, 
is  shown  by  Mr  Polya  by  the  example  of  the  function 

which  assumes  integral  values  for  all  integral  values  of  .r. 


64     Dr  Marr,  Submergence  and  cjlacial  climates  (luring  the 


Submergence  and  glacial  climates  daring  the  accumulation  of 
the  Cambridgeshire  Pleistocene  Dejwsits.  By  J.  E.  Marr,  Sc.D., 
F.R.S,  St  John's  College. 

[Read  5  February  1917.] 

A.     Introductory. 

The  sequence  of  events  during  palaeolithic  times  is  still  a 
subject  surrounded  by  much  uncertainty.  The  area  of  the  Great 
Ouse  Basin  is  one  in  which  considerable  light  has  already  been 
thrown  on  vexed  questions,  and  as  the  examination  of  the  area  is 
carried  out  in  greater  detail,  important  results  will  be  obtained, 
for  in  this  area  we  get  evidence  of  the  relationship  of  the  palaeo- 
lithic deposits  to  those  which  were  formed  during  a  period  of 
submergence  and  re-emergence,  and  also  to  accumulations  which 
give  evidence  of  the  occurrence  of  more  than  one  cold  period. 

The  general  distribution  of  the  palaeolithic  deposits  of  the 
district  around  Cambridge,  and  their  main  characters,  have  long 
been  known,  and  an  account  of  the  deposits,  with  references  to 
the  previous  literature,  is  given  in  the  Geological  Survey  Memoir 
The  Geology  of  the  Neighbourhood  of  Gambridqe,  published  in 
1881. 

Since  that  memoir  appeared,  further  light  has  been  thrown 
on  the  deposits,  especially  by  Professor  Hughes,  who  has  given  his 
latest  views  in  a  paper  entitled  Tlie  Gravels  of  East  Anglia 
(Cambridge  University  Press,  1916). 

I  have  devoted  much  attention  to  this  subject  during  the  last 
six  years  and  hope  to  describe  my  detailed  results  elsewhere. 
The  present  paper  is  concerned  with  a  discussion  of  the  main 
problems  involved,  in  hopes  that  it  may  direct  the  attention  of 
workers  to  the  importance  of  further  observations,  for  the  deposits 
with  which  we  are  concerned  are  only  exposed  temporarily  during 
the_  working  of  gravel-pits  and  the  digging  of  foundations  and 
drains,  and  it  is  desirable  that  all  temporary  excavations  should 
be  carefully  studied,  and  the  objects  obtained  rendered  available 
for  study  by  deposit  in  Museums,  for  isolated  specimens  in  private 
collections  are  usually  mere  objects  of  curiosity  devoid  of  scientific 
value. 

B.     Submergence  and  its  effects.     The  actual  sequence  of  deposits. 

In  the  fenland  and  on  its  borders  we  meet  with  marine  deposits 
above  sea-level,  which  have  long  been  known  around  March  and 
Narborough.      They  occur  above  and  below  fen-level  at  March 


accumulation  of  the  Gamhridyeshire  Pleistocene  Deposits     G5 

and  undoubted  marine  deposits  containing  sea-shells  are  found 
to  a  height  of  at  least  50  feet  above  sea-level  in  the  Nar  Valley, 
and  deposits  up  to  80  feet  above  sea-level  have  been  claimed  as 
marine.  Unfortunately  no  exposure  of  these  Nar  Valley  beds 
has  been  seen  for  a  very  long  time,  and  their  exact  upward  limit 
is  a  matter  which  must  remain  unsettled  until  new  excavations 
are  made.  It  is  held,  with  good  reason,  that  the  beds  of  March 
and  the  Nar  Valley  are  geologically  contemporaneous  in  the  sense 
that  they  belong  to  the  same  period  of  sea-invasion,  which  was 
subsequent  to  the  accumulation  of  the  chalky  Boulder  Clay ;  and 
as  there  is  good  evidence  that  much  of  the  fenland  was  low-lying 


Nar Level 


Fig.   1. 

AB.         Slope  of  ground  before  marine  gravels  were  deposited. 
CD.  ,,  „     after  ,,  ,,  ,. 

a.  Tract  of  marine  gravels. 

b.  ,,  interdigitating  marine  and  fluviatile  gravels. 

c.  ,,         fluviatile  deltaic  deposits. 

d.  ,,        erosion  in  valley  towards  its  head,  during  period  of  deposit  of 

a,  b,  c. 
1,  2,  3.     Order  of  formation  of  deposits  in  tracts  c  and  d  respectively.    (1  is  oldest.) 
Vertical  scale  greatly  exaggerated. 

ground  after  this  boulder-clay  was  formed,  it  would  appear  prob- 
able that  the  March  gravels  are  earlier  than  those  of  the  Nar 
Valley,  and  therefore  that  a  gradual  silting  up  of  a  bay  of  the 
sea  took  place,  until  the  sediments  reached  a  height  of  at  least 
50  feet  above  present  sea-level. 

During  this  period  of  silting  the  rivers  Ouse,  Cam  and  others 
would  build  delta-deposits  along  the  lower  parts  of  their  courses, 
with  interdigitation  of  marine  and  fluviatile  deposits  in  an  inter- 
mediate belt  of  ground  as  shewn  in  figure  1.  In  this  delta- 
material,  the  chronological  sequence  of  deposit  would  be  from 
below  upward,  as  shewn  by  1,  2  and  3  in  the  belt  c.  The  upper 
waters  of  the  rivers  would  still  be  eroding,  and  the  sequence 
would  be  from  above  downwards  (see  figs,  in  tract  d). 

After  submergence  had  ceased,  it  would  be  replaced  by  re- 
emergence,  as  shewn  by  the  erosion  of  the  rivers  to  their  present 


66     Dr  Marr,  Submergence  and  glacial  climates  during  the 

levels,  and  new  deposits  4,  5,  . . .  (not  shewn  in  the  diagram)  would 
be  banked  against  or  laid  down  upon  those  formed  during  the 
period  of  subsidence  and  general  accumulation  in  tracts  c  and  a. 
It  will  be  seen  therefore  that  relative  height  of  deposits  above 
the  present  river- level  is  not  in  itself  a  necessary  indication  of 


The  geological  surveyors  gave  the  following  classification  of 
the  Cam  gravels: 

r  Lowest  Terrace 
Gravels  of  the  Present  River  System  I  Intermediate  Terrace 

[Highest  Terrace 
Gravels  of  the  Ancient  River  System. 

I  shall  treat  of  three  of  these,  leaving  out  of  account  the  gravels 
of  the  Intermediate  Terrace,  which  I  have  not  studied  extensively 
owing  to  poor  and  infrequent  exposure  of  recent  years.  I  shall 
speak  of  the  gravels  of  the  '  Ancient  River  System '  as  the  Obser- 
vatory gravels,  those  of  the  highest  terrace  of  the  '  present  river 
system '  as  the  Barnwell  village  gravels,  and  those  of  the  lowest 
terrace  as  the  Barnwell  Station  gravels.  The  ages  of  these 
deposits  will  ultimately  be  accurately  determined  by  an  exami- 
nation of  the  fossil  evidence,  including  implements  of  human 
manufacture.  So  far,  the  evidence  of  this  kind  points  to  the 
Barnwell  village  deposits  being  of  two  ages,  the  older  formed 
during  the  period  of  delta-growth,  the  newer  during  the  period 
of  re-emergence  and  erosion.  At  the  end  of  the  period  of  delta- 
growth,  and  therefore  of  an  age  intermediate  between  those  of 
the  supposed  two  Barnwell  village  deposits,  I  would  place  the 
Observatory  gravel,  and  certain  loams,  to  be  referred  to  later, 
and  after  all  of  these,  the  Barnwell  Station  gravel  marking  the 
culmination  of  the  period  of  re-erosion,  for  there  is  evidence  of  a 
later  period  of  sinking  and  deposit  after  this  was  formed.  This 
succession  is  represented  in  Fig.  2,  which  shews  a  section  across 
the  Cam  valley  at  Cambridge,  before  the  edges  of  the  valley  sides 
had  been  destroyed  leaving  the  Observatory  gravels  as  a  ridge 
with  lower  ground  on  either  side. 

In  the  figure  the  terms  Upper,  Middle  and  Lower  Palaeolithic 
indicate  the  ages  of  the  various  gravels  as  inferred  by  me  from 
the  palaeontological  evidence.  I  am  using  the  term  Middle  Palaeo- 
lithic in  the  sense  in  which  it  was  used  by  Prof.  Sollas  in  the 
first  edition  of  Ancient  Hunters  as  equivalent  to  Mousterian. 
I  believe  therefore  that  the  older  Barnwell  village  gravel  is  pre- 
Mousterian,  that  of  the  Observatory  (in  part  at  any  rate)  Mou- 
sterian, and  the  newer  Barnwell  village  gravel  and  that  of  Barnwell 
Station  post-Mousterian,  the  former  being  of  earlier  date  than 
the  latter. 

Mr  Jukes-Browne,  in  an  essay  on  the  Post  Tertiary  Deposits  of 


■I 


accumulation  of  the  Camhrldgeslnre  Pleistocene  Deposits     67 

Cambridgeshire,  advocated  a  change  in  the  direction  of  the  rivers 
near  Cambridge  between  the  formation  of  the  Observatory  gravels, 
and  those  which  he  regarded  as  belonging  to  the  'present  river 
system.'  That  such  a  change  occurred  is  admitted,  but  the  evi- 
dence points  to  all  the  deposits  save  those  of  the  Barnwell  Station 
terrace  having  been  formed  before  the  river  diversion  occurred. 

I  may  now  pass  on  to  consider  briefly  the  palaeontological 
evidence  in  favour  of  the  order  of  age  indicated  above,  leaving 
details  for  a  future  paper. 

In  the  pits  of  Barnwell  village,  and  of  the  Milton  Road  near 
Chesterton,  loams  are  sometimes  exposed  at  the  base  of  the  over- 
lying gravels.  These  loams  contain  Corhiculaflaviinalis,  and  with 
it  are  associated  Unio  Wtoralis,  Belgrandia  margiiiata,  and  Hip- 
popotamus,    On    the    continent    this    is   recognised    as    an   early 


Fig.  2. 

Section  acroiss  Cam  N.  of  Cambridge,  with  higher  valley-slopes  restored. 
The  figures  shew  the  suggested  order  of  formation  of   the  deposits.     Cross- 
hatching  represents  modern  alluvium  of  Cam. 

5.     Barnwell  Station  gravels  (Upper  Palaeolithic  2). 

4.     Newer  Barnwell  village  gravels  (Upper  Palaeolithic  1). 

3.     Loams  of  Huntingdon  Koad  area. 

2.     Observatory  gravels  (Middle  Palaeolithic). 

1.     Older  Barnwell  village  gravel  and  loam  (Lower  Palaeolithic). 

Z= Buried  channel. 

Vertical  scale  greatly  exaggerated. 

palaeolithic  fauna  of  Chellean  or  pre-Chellean  date,  and  there 
seems  to  be  no  evidence  of  the  reappearance  of  this  fauna  at  a 
later  date. 

In  the  Geological  Magazine  for  1878  (p.  400)  Mr  A.  F.  Griffith 
described  the  occurrence  of  a  palaeolithic  implement  from  one  of 
the  Barnwell  pits.  A  cast  of  this  is  in  the  Sedgwick  Museum, 
and  it  appears  to  be  of  Chellean  type. 

Further  afield,  the  occurrence  of  similar  implements  at  or  near 
fen-level  in  Swaffham  and  Soham  fens,  and  at  West  Row  near 
Mildenhall,  and  at  Shrub  Hill  near  Feltwell,  indicates  that  rivers 
had  excavated  their  channels  to  fen-level  in  those  times. 

There  are  patches  of  gravel  between  the  higher  Chesterton 
terrace  which  corresponds  to  the  Barnwell  village  terrace  and  the 


68     Dr  Marr,  Suhmerc/ence  and  glacial  climates  dnrmg  the 

Observatory  level,  but  no  sections  are  now  seen  in  them,  so  we 
may  pass  on  to  the  Observatory  deposits.  In  these  shells  and 
mammalian  bones  are  very  rare,  though  the  former  have  been 
found  in  concretions,  indicating  that  they  once  lay  in  the  gravels, 
but  have  since  been  dissolved.  Implements  are  relatively  abun- 
dant, and  I  have  found  a  large  number  during  recent  years. 
Many  of  them  are  of  Chellean  type,  others  probably  Acheulean, 
but  there  are  a  large  number  of  Mousterian  type,  some  having 
the  facetted  platform  which,  as  shewn  by  M.  Commont,  came  into 
use  in  Northern  France  in  Mousterian  times.  It  may  be  noted 
that  the  implements  of  Mousterian  type  are  patinated  differently 
to  and  in  a  less  degree  than  those  of  Chellean  type,  and  I  regard 
the  two  series  as  of  distinct  ages.  Either  the  deposits,  which  are 
thick  and  varied  in  character,  are  of  two  dates,  or  implements 
of  different  ages  lying  upon  the  surface  were  washed  into  the 
deposits  contemporaneously.  This  can  only  be  settled  by  finding 
a  number  in  situ,  a  work  of  great  difficulty,  but  the  evidence  is 
in  favour  of  the  latter  view. 

I  may  note  that  when  a  valley  is  being  deepened  implements 
of  one  age  only  are  likely  to  lie  in  abundance  near  the  spot  where 
the  gravels  were  accumulating,  but  when  there  is  general  aggra- 
dation, the  highest  deposits  of  the  delta-growth  are  likely  to 
receive  washings  of  implements  of  various  ages  which  have  been 
lying  together,  at  or  near  the  surface.  In  any  case  the  age  of  the 
newest  gravel  of  a  terrace  will  be  determined  by  the  implements 
of  latest  date. 

Lying  on  this  gravel  in  channels  are  reddish  sandy  loams, 
which  must  have  spread  over  the  gravel,  but  have  since  been 
destroyed  by  erosion  except  where  so  preserved.  There  is  also 
a  deposit  of  somewhat  similar  loam  but  of  a  lighter  colour  flanking 
the  gravel  at  a  lower  level  on  either  side.  It  is  rarely  exposed, 
and  only  in  shallow  sections,  but  I  believe  it  may  be  of  the  same 
general  date  as  that  lying  on  the  gravel. 

No  relics  have  been  found  in  it,  though  two  implements  of 
possible  Upper  Palaeolithic  date  were  found  on  the  loam  when  ' 
draining  the  Christ's  Cricket  Grouod,  but  they  may  well  have  ■ 
been  surface  finds.  Many  other  surface  finds,  some  of  apparent  ' 
palaeolithic  type,  are  found  on  this  loam  belt,  and  will  be  referred  i 
to  later.  | 

Those  gravels  of  the  terraces  of  Barnwell  village  age,  which  | 
I  would  refer  to  a  date  later  than  that  of  the  Gorhicula  gravels,  I 
are  now  exposed  in  a  pit  near  the  Milton  Road  and  in  another  | 
on  the  Newmarket  Road  near  Elfleda  House,  2\  miles  from  j 
Cambridge.  These  contain  a  fauna  differing  from  the  Gorhicida  \ 
fauna,  and  including  the  mammoth,  woolly  rhinoceros,  horse  and 
red  deer,  the  horse  being  abundant. 


accumulation  of  the  Cambridgeshire  Pleistocene  Deposits     69 

Implements  are  scarce,  but  in  both  pits  I  have  found  some 
suggestion  of  upper  palaeolithic  forms,  and  in  each  pit  a  water- 
worn  pot-boiler  has  been  discovered. 

In  the  Barnwell  Station  pit  the  common  mammal  is  the  rein- 
deer, associated  with  the  mammoth,  tichorhine  rhinoceros  and 
horse.  In  the  Geological  Magazine  for  1916  (p.  339),  Miss  E.  W, 
Gardner  and  I  recorded  the  occurrence  of  an  arctic  flora  in  this 
deposit,  with  abundance  of  leaves  of  Betula  nana.  A  long  pre- 
liminary list  of  the  other  plants  which  indicate  arctic  conditions 
was  made  by  the  late  Mr  Clement  Reid,  F.R.S.,  but  has  not  yet 
been  published.  A  few  worked  flints  of  undeterminable  date 
have  been  found,  but  the  fauna  indicates  the  late  palaeolithic 
period,  and  the  late  date  of  these  deposits  seems  to  be  shewn  by 
the  fact  that  whereas  all  the  others  are  apparently  connected 
with  the  old  drainage  line  extending  from  Cambridge  to  Somers- 
ham,  these  are  almost  certainly  parallel 'to  the  present  course  of 
the  Cam :  they  appear  indeed  to  be  the  upper  portion  of  the 
deposits  filling  an  old  buried  channel  of  the  Cam,  evidence  for  the 
occurrence  of  which  is  borne  out  by  certain  observations  made  by 
Prof  Hughes  in  the  paper  to  which  reference  has  been  given. 


C.     Climatic  Changes. 

There  is  much  difference  of  opinion  as  regards  the  occurrence 
of  alternating  glacial  and  interglacial  periods  in  Pleistocene  times, 
and  it  would  seem  that  some  light  is  thrown  upon  this  question 
by  the  Cambridgeshire  deposits  and  those  of  adjoining  counties. 

I  take  the  prevalent  view  that  the  implement-bearing  deposits 
from  the  beginning  of  Chellean  times  post-date  the  period  of  the 
Chalky  Boulder  Clay,  though  others  hold  a  different  view,  but  as 
the  local  evidence  bearing  upon  this  question  has  already  been 
recorded  I  need  not  enlarge  upon  this  point. 

If  the  succession  as  outlined  above  be  correct  the  following 
climatic  changes  seem  to  have  occurred  after  the  cold  period 
marked  by  the  accumulation  of  the  Boulder  Clay  : 

(a)  A  warm  period  during  the  formation  of  the  Corbicula- 
bearing  strata.     Arguments  in  favour  of  this  are  well  known. 

(6)  A  cold  period  during  the  accumulation  of  the  Observatory 
gravels(?)  and  the  newer  loams.  No  evidence  of  this  has  been 
advanced  in  this  area,  and  a  few  remarks  are  necessary. 

The  fauna  of  the  Observatory  gravels  tells  us  nothing,  and 
the  loams  have  hitherto  furnished  no  organic  remains,  but  a 
widespread  development  of  loam  marks  the  Mousterian  period, 
and  N.W.  Europe  is  believed  to  have  been  subjected  to  a  cold 
climate  during  part  of  the  period. 


70     Dr  Marr,  Submergence  and  glacial  climates  during  the 

The  sections  recently  seen  near  Cambridge  tell  us  little,  but 
a  brickpit  in  stratified  loam  with  'race'  nodules  similar  to  those 
found  in  the  Cambridge  sections  has  long  been  worked  near  the 
railway  between  Longstanton  and  Swavesey.  It  contains  boulders, 
and  is  actually  mapped  as  boulder-clay.  A  somewhat  similar 
loam  with  boulders  at  High  Lodge  near  Mildenhall  has  long  been 
known  for  its  implements  of  Mousterian  type.  These  deposits 
are  at  an  elevation  just  below  that  of  the  highest  palaeolithic 
gravels,  as  are  those  of  Cambridge. 

Further  afield  there  is  the  very  significant  section  at  Hoxne, 
described  in  detail  in  a  paper  drawn  up  by  the  late  Clement  Reid, 
F.R.S..  and  published  in  the  Report  of  the  British  Association  for 
1896.  ' 

At  that  locality  we  have  a  stratigraphical  sequence.  Above 
the  boulder-clay  lies  an  aquatic  deposit  marked  by  a  temperate 
fauna.  It  is  succeeded  by  loams  with  an  arctic  flora,  and  above 
that  are  loams  with  palaeolithic  implements.  They  have  been 
usually  regarded  as  Acheulean,  but  there  is  one  specimen  in  the 
Sedgwick  Museum  which  is  of  a  distinct  Mousterian  type.  Taking 
these  facts  into  consideration,  a  period  of  cold  climate  in  this 
country  in  Mousterian  times  seems  probable.  In  any  case,  the 
evidence  points  to  a  difference  of  date  of  the  arctic  plant-beds  of 
Hoxne  and  Barnwell  Station. 

(c)  The  fauna  of  the   beds   of  the   Barnwell   village   terrace 
claimed  here  as  of  newer  date  than  those  containing  CoriicM^ajj 
suggests  an  amelioration  of  the  climate,  but  in  the  absence  of  a  '' 
well  preserved  flora,  this  is  doubtful. 

(d)  The  Barnwell  Station  flora,  as  before  observed,  is  distinctly 
arctic,  and  when  this  flora  lived  here,  we  can  hardly  suppose  that 
our  higher  hills  escaped  glaciation.  The  same  remark  may  be 
made  of  the  Hoxne  flora. 

This  series  of  changes  would  accord  with  the  classification  of 
the  beds  on  the  continent  thus  : 

European  Continent  Cambridgeshire 

Pleistocene 

Wiirm  glaciattoxi  Barnwell  Station  beds. 

Waiiii  period  Newer  Barnwell  village  deposits. 

Riss  glaciation  Observatory  gravels  and  loams. 

Warm  period  Corhicula  gravels. 

Mindel  glaciation  Chalky  Boulder  Clay. 

Pliocene 
Warm  period  Cromer  'Forest'  series. 

Giinz  glaciation  Chillesford  beds. 

I  merely  put  this  forward  tentatively,  claiming  however  that 
we  have  in  Cambridge  proofs  of  two  if  not  three  Pleistocene  cold 
periods. 


accumulation  of  the  Cambridgeshire  Pleistocene  Deposits     71 

D.     Surface  Implements. 

Implements  of  all  ages  from  earlier  palaeolithic  to  recent 
times  are  found  lying  together  on  the  surftice.  Some  no  doubt 
have  got  there  from  the  erosion  of  deposits  which  contained  them, 
others  belong  to  the  surfece.  My  object  is  to  insist  on  their 
careful  collection,  with  exact  records  of  their  localities,  even  to 
the  particular  position  in  a  field  where  thej^  lay. 

If  they  can  be  shewn  to  be  limited  to  heights  above  those  of 
a  particular  deposit,  they  may  yield  valuable  information  as  to 
geological  changes. 

Two  areas  in  which  surface  implements  are  abundant  are 
found  very  near  Cambridge,  one  on  the  tract  between  Castle  End 
and  Girton  on  either  side  of  the  Huntingdon  Road,  on  the  ground 
occupied  by  the  Observatory  gravels  and  loams,  the  other  a  little 
south  of  Fen  Ditton,  between  the  railway  and  the  river,  and  at 
no  great  height  above  the  latter.  They  have  not  been  yet 
sufficiently  studied  to  enable  one  to  draw  definite  conclusions,  but 
the  former  group  does  not  seem  to  occur  below  the  level  of  the 
Barnwell  village  terrace,  which  suggests  that  the  river  may  have 
eroded  its  valley  below  that  level  to  its  present  position  since 
those  implements  were  made.  The  other  set  marks  the  position 
of  a  site  on  a  terrace,  which  is  I  believe  the  terrace  of  the 
Barnwell  Station  deposits,  and  would  indicate  the  formation  of 
that  terrace  before  this  set  of  implements  was  manufactured. 

As  the  above  is  merely  a  preliminary  account  of  these  deposits, 
I  have  not  burdened  it  with  references,  nor  have  I  acknowledged 
the  many  friends  who  have  helped  in  the  collection  of  implements 
and  other  objects. 

The  bulk  of  the  implements  on  which  my  conclusions  are 
based  were  collected  by  myself,  and  the  rest  by  friends  chiefly 
under  my  supervision,  and  in  no  case  has  any  implement  been 
purchased  from  workmen,  so  that  the  collection,  which  will  be 
deposited  in  the  Sedgwick  Museum,  is  of  value,  inasmuch  as  each 
implement  is  known  to  have  been  obtained  from  the  locality 
assigned  to  it. 


72     Mr   Weatherburn,  On  the  Hydrodynamics  of  Relativity 


On  the  Hydrodynamics  of  Relativity.  By  C.  E.  Weather- 
burn,  M.A.  (Camb.),  D.Sc.  (Sydney),  Ormond  College,  Parkville, 
Melbourne. 

{Received  15  December  1916 :  read  5  February  1917.] 

I.    The  Equations  of  Motion. 

I  1.  Relativistic  equations  for  the  adiabatic  motion  of  a 
frictionless  fluid  have  been  found  by  Lamia*  and  Lauef  in  the 
form 

dt^     ■'         dx^     ■^         dy^  dz^     '      y  dx 

^  7\  7)  f)  ^  7)  P 

—  (kv)  +  11;^  (kv)  +  V  ;^-  (kv)  +  iv  X-  (kv)  +  -  ^  =  Fy...(l); 

where  m,  y,  w  are  the  components  of  velocity  at  the  point  {x,  y,  z) 
relative  to  a  definite  system  of  reference  8 ;  X,  Y,  Z  those  of  the 
impressed  force  per  unit  of  normal  rest-mass ;  and 

ry=  ^  (2), 

Vc^  —  {u^  •\-v^  +  id^) 

c  being  the  constant  velocity  of  light.  The  significance  of  the 
symbols  P  and  k  is  as  follows. 

Since  the  motion  is  adiabatic  the  rest-mass  of  an  element  of 
fluid  is  determined  by  one  variable  only,  say  the  pressure  p. 

If  we  choose  some  definite  pressure  p^  as  the  normal  or 
standard  pressure,  the  element  has  a  definite  constant  normal 
rest-mass  hm^.  If  the  element  occupies  a  volume  hV  relative  to 
the  system  of  reference  8,  the  density  k  relative  to  that  system  is  ^| 
defined  by 

,  _8mo 

*  Ann.  der  Physik,  Vol.  37,  p.  772  (1912). 

•|-  Das  Relativitdtsprinzip,  §  36  (2nd  ed.  1913).  For  a  more  general  discussion  of 
the  mechanics  of  deformable  bodies  from  the  standpoint  of  Relativity,  cf.  Herglotz, 
Ann.  der  Physik,  Vol.  36,  p.  493  (1911);  also  a  paper  by  Igndtowsky,  FIn/s.  Zeit., 
Vol.  12,  p.  441  (1911). 


I 


Mr   Weathei'burii,  On  the  Hydrodynamics  of  Relativity     73 

Using  a  dash  t(j  refer  in  every  case  to  the  rest-systeui  tS",  we 
have  for  the  rest-density 

37»o  _  hm^  _  k 
~BV'.~y8V      y   ^"^• 

The  function  F  is  defined  by  the  integral 

i'=rt (4), 

•'  Pa  f^ 

and  in  terms  of  this  function  k  is  given  by 

-^=^(1  +  ^) <'''■ 

For  the  rest-system  *S"  the  quantity  y  has  the  value   unity, 
while  K  becomes 

k'=1+-^ (5'). 

The  constancy  of  normal  i-est-mass  leads,  as  in  the  classical 
theory,  to  an  equation  of  continuity 

¥  +  fl.<'-«>  +  a^<''">  +  3i*^^''>  =  ° <">■ 

§  2.     Using  F  and  v  for  the  force  and  velocity  vectors,   we 
may  write  the  equations  of  motion  more  conveniently 

|(«v)  +  v.V(/cv)-[--VP=F (7). 

ot  y 

Then  because  the  gradient  of  the  scalar  product  of  two  vectors  is 
given  by 

V  (a  •  b)  =  b  •  Va  +  a  •  Vb  -f-  b  x  curl  a  +  a  x  curl  b, 
the  second  term  of  (7)  is  equivalent  to 

-    V  (k-v")  -  V  X  curl  (kv), 
Ik 

while,  in  virtue  of  (5'),  VP  =  c'-'^/c'.     Hence  the  equation  may  be 
exjjressed  in  the  form 

r)  1 

^^  («v)  -I-  „    V  {k-y-  -I-  cV'-)  -  V  X  curl  («v)  =  F. 
ot  'Ik 

But  again  the  second  term  is  equal  to 
1 


VOL.  XIX.  PARTS  II.,  m.  6 


2.^ 


74     ifr   Weatherburn,  On  the  Hydrodynamics  of  Relativity 
and  the  equation  of  motion  takes  the  ver}^  convenient  form 

^(a:v)  + c-V/c  +  2w  X  v=  F   (8), 

where  we  have  written 

2w  =  curl  (kv). 

In  cases  where  the  impressed  force  F  admits  a  potential,  so 
that  F  =  —  V  F,  our  equation  reduces  to 

^(«v)  +  V(t;-/c+F)  +  2wxv  =  0 (8'). 

§  3.  Glehsch's  transformation^.  The  equation  of  motion  may 
be  expressed  in  terms  of  functions  analogous  to  those  of  Clebsch 
if  we  write 

KV  =  V(f)  +  A.V/X (9), 

(ji,  \,  fi  being  three  independent  functions  of  x,  ?/,  z  and  t.  Taking 
the  curl  of  both  members  we  find  immediately  that 

2w  =  VXx  V;^     ^0). 

The  function  w  =  i  curl  (kv)  plays  the  same  part  in  the  present 
analysis  as  |-  curl  v  in  classical  hydrodynamics.  It  will  therefore,  by 
analogy,  be  called  the  vorticity ;  and  a  line  whose  direction  at  any 
point  IS  the  direction  of  w  at  that  point,  a  vertex  line.  Since 
by  (10)  w  is  perpendicular  to  both  VX  and  V/x  it  is  clear  that  the 
vortex  lines  are  the  intersections  of  the  surfaces 

A,  =  const.,     /J,  =  const. 

Using  then  dots  to  denote  partial  differentiation  with  respect 
to  t,  and  assuming  the  existence  of  a  force  potential,  we  may  write 
(8')  as  -^ 

-  V  ( F+  c-a:)  =  V(j)  +  X/i)  +  XV/i  -  ^Vx 

+  (v .  VX)  VyLi  -  (V  .  V/x)  VX 

which  may  be  neatly  expressed  in  the  form 

§v,-|vx  +  Vi.=  o  (u), 

where  the  function  H  is  given  by  the  equation 

If=  (f)  +  \jiL  +  V+C"K    (12). 

_  *  ^f;  Basset,  Treatise  on  Hydrodytiamics,  Vol.  1,  p.  28  ;  also  Silberstcin,  Vectorial 
Meciianics,  p.  146. 


Mr   Weatherburn,  On  the  Hydro(hjnamics  of  Relativity     75 

On   scalar   multiplication  of  (11)  by  w,   it   follows  in  virtue 
of  (10)  that 

w.V//  =  0, 

showing-  that  H  is  constant  along  a  vortex  line.  It  can  also  be 
shown  that  H  is  independent  of  x,  y,  z  and  is  therefore  a  function 
of  i  only.     For  taking  the  curl  of  (11)  we  deduce 

On  scalar  multiplication  by  Vx  it  follows,  by  (10),  that 
and  similarly  that 


w.Vl'^fl^O, 


From  these  we  deduce  as  in  the  old  theory*  that 

^  =  ^  =  0   ...(13). 

dt      dt  ^ 

Thus  the  first  two  terms  disappeai-  from  (11),  which  becomes 
simply  VH  =  0,  showing  that  H  is  constant  in  space  and  is 
therefore  a  function  of  t  only ;   or 

(j)  +  \/l  +  V  +  c^fc  =  H  (t)     (14). 

From  (13)  it  is  clear  that  the  surfaces  \  =  const,  and  /x  =  const., 
and  therefore  also  the  vortex  lines  which  are  their  lines  of  inter- 
section, are  always  composed  of  the  same  particles  of  fluid. 

§  4.  Steady  motion.  When  the  motion  is  steady  partial 
derivatives  with  respect  to  t  are  zero.  If  then  the  impressed 
force  is  derivable  from  a  potential  V,  (8')  becomes 

2v  X  w  =  V  ( K  +  c-k)    (15), 

and  the  equation  of  continuity 

div(/.v)  =  0    (16). 

If  we  multiply  (15)  scalarly  by  v  the  first  member  vanishes, 
showing  that 

vV(  l^  +  c-/^)  =  0. 

Thus  the  function  V  +  c-k  is  constant  along  a  line  of  flow. 
Similarly  scalar  multiplication  of  (15)  by  w  gives 

w.V(F+c-^/c)  =  0, 
*  Cf.  Biis-et,  lor.  cit.  p.  29. 


76     Mr   Weatherhurn,  On  the  Hijdrodynamics  of  Relativity 

and  therefore  V  +  C'k  is  constant  also  along  a  vortex  line.  This 
is  a  particular  case  of  the  more  general  theorem,  proved  in  the 
preceding  section,  that  H  is  constant  along  a  vortex  line.  Thus 
the  surface 

V  +  c'-V  =  const, 
is  composed  of  a  double  system  of  vortex  lines  and  lines  of  flow. 

II.     Irrotational  Motion. 

§  5.  When  the  vorticity  i  curl  («v)  is  zero  the  motion  will  be 
termed  irrotational  or  non-vortical,  being  analogous  to  the  motion 
of  that  name  in  the  older  theory.  In  this  case  «v  can  be  expressed 
as  the  gradient  of  a  scalar  function  0,,  which  may  be  called  the 
velocity  potential :  i.e. 

'cv  =  V(f} : (17). 

The  lines  of  flow  are  orthogonal  to  the  surfaces  of  equal  velocity 
potential. 

The  equation  of  motion  can  always  be  integrated  when  a  force 
and  a  velocity  potential  exist.     For  (8')  then  becomes 
V  (<^  +  c'k  +V)  =  0. 

The  function  in  brackets  is  therefore  constant  throughout  the 
liquid,  and  will  be  a  function  of  t  only;    i.e. 

4>  +  c'K+V=f(t) (18). 

This  is  the  required  integral  of  the  equation  of  motion.  An 
arbitrary  function  of  t  may,  however,  be  incorporated  in  the 
velocity  potential  cf),  and  this  equation  then  written  without  loss 
of  generality 

(f)  +  c-K+  F=0    (18'y 

When  the  irrotational  motion  is  steady  (c'^k  +  V)  is  constant 
throughout  the  liquid,  and  is  also  invariable  in  time.  In  the 
preceding  section,  where  w  was  not  assumed  to  be  zero  this 
function  was  only  proved  constant  along  vortex  lines  and  lines  of 
flow. 

The  equation  of  continuity  (6),  or  as  it  may  be  written 

dk      ,    -. 

^  +  ^-divv  =  0, 

may  be  expressed  in  terms  of  0,  if  we  write  kvJk  for  v,  and  expand 
the  divergence  of  the  quotient.     The  equation  then  becomes 

|logA-  +  v(l).Vc/,  +  lv^0=.O  (19). 


Mr   Weatherhurn,  On  the  Hydrodynamics  of  Relativity     77 

This  form  is  not  so  short  as  in  the  ordinary  theory,  nor  can  we 
obtain  Laplace's  equation,  as  there,  by  assuming  the  Huid  incom- 
pressible, for  such  an  assumption  is  inconsistent  with  the  theory 
of  relativity*. 

§  6.  Steadily  rotating  fluid.  Suppose  that  the  Huid  is  in 
a  state  of  steady  rotation  about  the  ^^-axis,  and  that  the  angular 
velocity  of  rotation  O  is  a  function  of  the  distance  r  from  that 
axis.  We  shall  now  determine  what  must  be  the  form  of  this 
function  in  order  that  a  velocity  potential  may  existf-  If  i,  J:  k 
are  unit  vectors  in  the  directions  of  the  coordinate  axes 

V  =  rn. 
For  irrotational  motion  this  velocity  must  satisfy  the  equation 

curl  {kv)  =  0, 

that  is  IkH  +  r  V  ('<:^)  =  0, 

dr 

the  integral  of  which  is 

/tfir-  =  const.  =  fjb, 

say,  so  that  /cH  — ^     (A). 

The  velocity  potential  <^  is  then  given  by 

dd)      .         1  d<h  it 

dr  r  do  r 

showing  that  (f)  =  fid  +  const (B), 

which  is  an  example  of  a  cyclic  velocity  potential.  The  integral 
of  the  equation  of  motion  is  by  (18') 

c"k  +  V=0 (C). 

But  K  involves  v"  and  therefore  O,  which  is  itself  expressed  in 
terms  of  k  by  (A).     This  equation  however  gives 

K'  K-C^ 

whence  12-  =  -„ ,   „  „  ,„  , 

r-{/jb'  +  r''c-K-) 

*  Cf.  §  10  below.  It  will  be  shown,  howeyer,  in  §  11  that  V-</)  =  0  is  the 
equation  of  continuity  for  the  steady  irrotational  motion  of  a  Huid  of  minimum 
compressibility. 

t  Cf.  Lamb,  Hydrodynamics,  §  28  (1st  ed.). 


78     Mr   Weatherhurn,   On  the  Hydrodjjnarnics  of  Relatiinty 

K  being  given  by  (5').     On  substitution  of  this  vabio  in  (C)  the 
integral  of  the  equation  of  motion,  viz. 

becomes  V  +  ~  \l uC^  +  r-c-K.'- =  0   (D). 

§  7.  FloLV  and  circulation.  We  define  the  flow  from  a  point  P 
to  another  Q,  along  a  path  of  which  ds  denotes  an  element,  as  the 
quantity 

[Q 

kV  •  ds. 
J  p 

Whenever  a  velocity  potential  exists  this  is  equal  to  <^y  —  (j)^.    The 
circulation  round  a  closed  curve  is  the  line  integral 


/=     /cv.f/s  (20) 


taken  round  that  closed  curve.     This,  by  Stokes'  theorem,  is  equal 
to  the  surface  integral 


7  =  1" curl («v).nrf;S'    (20') 


taken  over  any  surface  drawn  in  the  region  and  bounded  by  the 
closed  curve.  When  the  motion  is  irrotational  the  integrand  is 
zero,  and  the  circulation  round  the  closed  curve  vanishes.  It 
follows  that,  for  a  simply-connected  region,  the  velocity  potential 
is  single- valued. 


III.     Vortex  Motion. 

I  8.  When  the  vorticity  w  is  not  zero  the  motion  will  be 
called  vortical  or  vortex  motion.  A  vortex  tube  is  one  bounded  bj 
vortex  lines.  Considering  the  portion  of  a  vortex  tube  betAveei 
any  two  cross  sections,  we  find  as  usual  on  equating  the  volume 
and  surface  integrals 

0  =     div  curl  (/cv)  dr  =  I    2w  •  nd,S, 

that  the  moment  of  the  vortex  tube   1  vj'XidS,  Avhere  the  inte 

gration  is  extended  over  the  cross  section,  is  the  same  for  all 
sections.  And  hence,  as  in  the  classical  theory,  the  vortex  lines 
either  form  closed  lines,  or  else  end  in  the  surface  of  the  fluid. 


Mr   Weatherbarn,  On  the  Hydrodijnaniics  of  Relativitij     79 


I  shall  now  show  that,  on  the  assumption  of  a  force  potential, 
Kelvin's  theorem*  of  the  constancy  of  the  circulation  in  a  closed 
filament  moving  with  the  fluid  is  true  in  the  present  case  also. 
Consider  a  closed  filament  consisting  always  of  the  same  particles, 
and  let  ds  be  a  vector  element  of  its  length  and  ds  the  correspond- 
ing scalar.     Then  the  circulation  round  it  is 


/  =      kv  •ds. 


The  time  rate  of  change  of  this  is 

dl 

dt  ' 


|(.v).r/s+.v.(;jj 


f/s 


VF--VP  +  /cv.(f/s.V)v 

7 


--F+/CV.  _- 

OS  OS 


ds 


ds 


"lv" 


.(21). 


Now  the  last  integral  is 


ds 


c-  c 

,  V/c  — 


(c"- 


^.  die     K  8y^ ,   , 

''hs'-^.Ts^^'- 


T  27  Vc2  -  V- 

On  substitution  of  this  value  in  (21)  that  equation  reduces  to 


dr 

dt 


9«: 


9^     9  /     n' 


96'       ds      ds 


ds. 


Hence,  since  the  path  of  integration  is  closed  and  k,  V,  and  kv^ 
are  single-valued  functions,  the  integral  vanishes,  showing  that 


f- 


.(22). 


Thus  the  circulation  does  not  alter  with  the  tiipe. 

Corollary.  If  /  is  zero  at  any  instant  it  will  remain  zero.  In 
particular,  if  the  motion  is  irrotational  at  any  instant  it  will  remain 
so,  provided  that  the  impressed  forces  have  a  potential. 

§  9.  Helmholtz's  theoremsf.  That  these  theorems  are  true  in 
the  present  theory  also  follows  without  difficulty  from  the  form  (8') 
of  the  equation  of  motion.  For  taking  the  curl  of  both  members 
we  have 


dw 

'dt 


+  curl  (w  X  v)  =  0. 


*  Of.  Silberstein,  loc.  cit.  p.  161,  for  the  proof  of  the  ordinary  theorem. 
t  Ibid.  Y>p.  163 — 65. 


80     Mr   Weatherburn,  On  the  Hydrodynamics  of  Relativity 

Expanding  the  second  term  and  using  the  equation  of  con- 
tinuity, we  find 

rfw      T^  dk  _ 

dt       k  dt  ~    ' 

which,  after  division  by  k,  may  be  written 

d  fVT\       w    ^ 

s(l)=I-^^  ■; (23). 

Differentiation  with  respect  to  t  gives 

cZ-  /w\      fd  Mv\    -.        w     ;'d  _ 

If  then  w  vanishes  at  any  instant  it  follows  from  (23)  that  the 
first  derivative  of  w/Z;  also  vanishes,  and  from  the  next  equation 
likewise  the  second  derivative  at  that  instant.  Similarly  all  the 
derivatives  with  respect  to  t  vanish  at  that  instant,  and  the 
quantity  -wjk  remains  permanently  zero,  so  that  the  motion  con- 
tinues irrotationa.l. 

Further,  the  moment  of  a  vortex  filament  does  not  vary  with  the 
time.  For  if  ds  is  an  element  of  such  a  filament  moving  with  the 
fluid 

ds  =  wds/w, 

and  -J-  (ds)  =  ds*  Vv  =  —  w  •  Vv, 

at  w 

so  that  (23)  is  equivalent  to 

d  /wX        w   d  ,  -,  ^ 

<sUJ  =  arf(('^'>    (24). 

Now  if  jj,  is  the  moment  of  the  filament,  dm^  the  constant 
normal  rest-mass  of  the  element  considered,  and  a  the  cross- 
sectional  area 

fx  =  aw,    dniQ  =  kads, 
,1    ,                                       w          ds 
^°"'''*  k  =  ^d^.   (25)- 

Substituting  this  value  in  (24),  and  remembering  that  dm^  is 
constant,  we  have 

|(/.^s)  =  ;x|(rfs), 

and  therefore  -^  =  0, 

dt 

showing  that  the  moment  of  the  filament  remains  constant. 


3h^   Weatherhurn,   On  the  Hydrodynamics  of  Relativity     81 

It  has  been  proved  already  that  a  mrtex  filament  consists 
alvmys  of  the  same  particles  of  fluid,  though  this  can  also  be  now 
deduced  from  (24)  and  (25),  using  the  invariability  of  yu.. 

IV.     Fluid  of  Minimum  Compressibility*. 

§  10.  According  to  the  theory  of  Relativity  no  velocity  can 
exceed  that  of  light.  Hence  there  is  no  such  thing  as  an  incom- 
pressible fluid ;  for  such  a  fluid  would  admit  a  wave  propagation 
with  infinite  velocity.  A  fluid  of  minimum  compressibility  is  one 
in  which  a  wave  can  attain  a  velocity  equal  to  that  of  light ;  and 
for  such  a  fluid  the  quantity  k  is  directly  proportional  to  the 
densityf 

K  =  k/kJ,     K^k'jk;     (27), 

where  A:,,'  is  a  constant  representing  the  normal  rest-density,  i.e.  the 
rest-density  corresponding  to  the  normal  pressure  p^. 

For  a  fluid  of  minimum  compressibility  the  equations  of  motion, 
energy  and  continuity  may  by  (27)  be  expressed  in  terms  of  the 
velocity  v  and  the  rest-density  k'.     The  equation  of  motion,  viz. 

becomes  on  substitution 

,  dv         dk     c'-„,,      J  ,„ 
^'  -ZtT  +  V  ,^  +  -  V^'  =  k,'F. 
dt  at      y 

Dividing  by  7  and  using  the  equation  of  continuity  to  transform 
the  second  term,  we  have  at  once 

k'  (^  -  V div  v)  +  (c^  -  v-^)  Vk'  =  A-o'F/7    (28), 

which  is  the  equation  of  motion  in  the  required  form. 

Multiplying  this  equation  scalarly  by  v,  and  transfortning 
V  •  V/t',  we  obtain 

,,/lrfv^        ,  ,.       \      ,,        ,,  fdk'      dk'\      /co'F.v 

.11.'  .1      //.  a/^J  _  ^,2\ 

Now 


dk' 

_d  fk  Vc-  - 

-"') 

dt 

~  dt\        c 

_  dk  Vc^  -  V-' 

k 

dv^' 

dt        c 

2c  "Jc^  -  v" 

dt 

=  —  k'  div  V  - 

1  y-k'  dv- 

2  c-    dt  ' 

*  Latnla,  loc.  cit.  p.  788 ;  Laue,  loc.  cit.  §  37.  f  Laue,  loc.  cit.  p.  241. 


82     Mr    Weatherburn,  On  the  Hydrodijnaribics  of  Relativity 

in  virtue  of  the  equation  of  continuit}-.  On  substitution  of  this 
value  in  the  last  equation  it  becomes  simply 

7/1-  C"    V"    U/C  Wn       __  //-»/,N 

//divv+        ^     ~-  =  --^F.v  (29), 

c-       ot  C^J 

which  is  the  energy  equation  in  terms  of  k'  and  v.  These  equations 
(28)  and  (29)  are  identical  with  those  found  otherwise  by  Lamia* 
and  Lauef.     The  equation  of  continuity  is  as  before 

^"  +  ^divv-0    (30), 

which  takes  the  required  form  if  k  is  replaced  by  jk'. 

§  11.  Steady  irrotational  motion.  In  virtue  of  (27)  the 
equation  of  continuity  may  also  be  written 

— +  div(/cv)  =  0  (81), 

Ob 

and  therefore  when  the  motion  is  irrotational 

|  +  ^^^  =  0 (31'). 

If  it  is  also  steady  the  iirst  term  is  zero,  and  we  have  (as  in  the 
older  theory  for  the  case  of  an  incompressible  fluid) 

^^(/)  =  0 (31"). 

Thus  for  steady  irrotational  motion  of  a  fluid  of  niinimimi  com- 
pressibility the  velocity  jjotential  satisfies  Laplace  s  equation. 

It  follows  immediately  that  for  such  a  fluid,  filling  a  simply- 
connected  region  within  a  hollow  shell,  which  is  fixed  relative  to 
some  system  of  reference  S,  steady  irrotational  motion  relative  to 
that  system  is  impossible.     For  by  Green's  theorem 

.K'V'dT  =  I  {^(py^dr  =  —     (f)/<:v  •  ndS  —  I  (f)V-(f)dT. 

Now  the  last  integral  vanishes  by  the  equation  of  continuity. 
So  also  does  the  last  but  one  :  for  v  •  n  is  zero,  being  the  normal 
velocity  at  the  surface  of  the  fluid.     Hence 


/ 


fc-v'-dr  —  0, 
showing  that  v  must  vanish  identically  throughout  the  fluid. 

*  Loc.  cit.  p.  792. 
•  t  Loc.  cit.  p.  244. 


Mr   WeatJierbuni,  On  the  Hydrodij navvies  of  Relativity     83 

In  the  present  case*  the  integral  of  the  equation  of  motion 
found  in  §  5,  viz. 

takes  the  form 

c^/, +  /,;'F=0, 

or,  in  terms  of  the  rest-densit}^  //, 

cH-'  +  Vk\;\/c--v'  =  0. 

§12.  l^^teady  motion  in  two  dimensions.  Supposing  the  fluid 
of  minimum  compressibility,  let  its  steady  motion  be  parallel  to 
one  plane — the  plane  xy.  Introduce  a  function  y^  satisfying  the 
relations 

dy\ 


dyjr 
KV  =        ^ 

on; 


•(32), 


u,  V  being,  as  in  §  1,  the  components  of  velocity  parallel  to  the 
axes  of  a;  and  y  resjjectively.  Such  a  function  -v/r  exists,  the 
equation  of  continuity 

div  («v)  =  0 

being  satisfied  identically.  The  function  yfr  is  proportional  to  the 
flux  of  matter  across  a  line  AP  drawn  from  a  fixed  point  A  to  the 
variable  point  P  (x,  y).  For  owing  to  an  infinitesimal  displacement 
Sx  of  F  the  increment  in  the  flux  of  matter  is 

kvSx  =  ku'icvSx  =  k,'  ~  Sx. 

ox 

Thus  if  ^  denote  the  flux 

-,^  o.i:  =  /in    :  -  dx. 

ox  ox 

Similarly  Jy  ^^  ^  ^'"  Jy  ^^' 

showing  that  '^P  =  kj-yjr, 

as  stated.  The  part  played  by  this  function  yjr  is  exactly  similar 
to  that  of  the  stream  function  in  the  two-dimensional  motion  of  a 
liquid  in  the  classical  theory.  The  present  function  also  is  a  true 
stream  function.  Its  value  is  independent  of  the  path  chosen  from 
-4  to  P  provided  the  region  is  simply-connected.     For,  if  ^4PP  and 

*  Lamia  considers  only  the  case  of  free  motion  (F=  const.) ;  loc.  cit.  p.  71*5. 


84     Mr   Weatherbarn,  On  the  Hydrodyncmdcs  of  Relativity 

ACP  are  two  different  paths,  the  flux  across  the  complete  boundary 
ACPBA  is 


h-v  .  ndfi  —  I  div  (7bV)  dr  =  0, 


as  is  also  obvious  because  the  motion  is  steady.  The  lines 
■y^r  =  const,  are  the  actual  stream  lines :  for  if  P  moves  subject 
to  this  condition  there  is  no  flux  across  the  path  traced  out  by 
that  point. 

The  above  is  true  whether  the  motion  is  irrotational  or  vortical. 
The  vorticity  w  is  equal  to 


,(34). 


and  therefore  for  irrotational  motion  -^  must  satisfy  Laplace's 
equation 

V^f  =  0 (33). 

If  this  relation  is  satisfied  there  is  a  velocity  potential  (f),  and  (32) 
may  then  be  expressed  in  the  form 

•    dcf)  _       d-\fr  I 

dx  dy  I 

dy  dx ) 

These  are  identical  with  the  relations  subsisting  between  the 
stream  function  and  the  velocity  potential  in  the  classical  theory 
of  the  two-dimensional  irrotational  motion  of  a  liquid.  They  are 
the  conditions  that  (jy  +  iyfr  should  be  a  function  of  the  complex 
variable  x  +  iy.  The  theory  of  such  functions  may  then  be  used 
as  in  the  theory  referred  to*,  to  give  various  possible  forms  of 
stream  lines  and  lines  of  equal  velocity  potential. 

§  13.  Source,  sink  and  doublet.  Similarly  the  irrotational 
motion  of  a  fluid  of  minimum  compressibility  defined  by  the 
velocity  potential 

/-^■l («^>' 

where  r  is  the  distance  from  a  fixed  point  0,  corresponds  to  the 
assumption  of  a  continual  creation  of  matter  at  the  point  0, 
of  amount  4<7rm  per  unit  time.     For 


so  that  ^•v  =  —  . 

r^ 

*  Cf.  Lamb,  loc.  cit.  chap.  iv. 


i 


Mr   Weatherbarn,  On  the  Hydrodynamics    of  Relativity     85 

The  velocity  is  therefore  radial  from  0,  and  kv  is  inversely 
proportional  to  ?'^.  The  flow  of  matter  per  unit  time  across  the 
surface  of  a  sphere  of  radius  r  is  ^irm,  equal  to  the  rate  of  creation 
of  matter  at  0.  Such  a  motion  is  then  that  due  to  a  source 
of  strength  m  at  the  point  0.  If  the  negative  sign  in  (o5)  were 
replaced  by  a  positive  one,  we  should  have  the  motion  due  to 
a  sink  at  0  of  strength  ni.  And  finally  the  velocity  potential 
representing  a  doublet  at  0  of  moment  M  and  with  its  axis  along 
the  unit  vector  n  is 


*=""-(;) 


86  Mr  Hardy,  On  the  convergence 


On  the  convergence  of  certain  multiple  series.  By  G.  H.  Hardy, 
M.A.,  Trinity  College. 

[Received  15  May  1917.] 

1.  In  a  paper  published  in  1903  in  the  Proceedings  of  the 
London  Mathematical  Society*,  and  bearing  the  same  title  as  this 
one,  I  proved  a  theorem  concerning  the  convergence  of  multiple 
series,  of  the  type 

which  is  given  (with  an  improvement  in  the  conditions)  on  p.  89 
of  Dr  Bromwich's  Theory  of  infinite  series.  This  theorem  is  one 
of  a  class  of  some  importance ;  and  I  propose  now  to  state  and 
prove  the  leading  theorems  of  this  class  in  a  form  more  systematic 
and  general  than  has  been  given  to  them  before.  I  shall  begin  by 
recapitulating,  with  certain  changes  of  form,  some  known  theorems 
concerning  simply  infinite  series ;  and  I  shall  then  obtain  the 
corresponding  theorems  for  double  series  in  a  form  as  closely 
analogous  as  possible.  The  generalisation  from  double  series  to 
multiple  series  of  any  order  may  well  be  left  to  the  reader.    • 

Simply  infinite  series, 

2.  I  shall  say  that  a  function  a.,„,,  real  or  complex,  of  a  positive 
integral  variable  m  is  of  hounded  variation  if 

■^  i  (^m  ~  f'-m+l  I 
1 

is  convergent.  It  is  plain  that  this  condition  involves  the  existence 
of  a  =  lim  a^n- 

Theorem  1.  The  necessai^y  and  sufficient  condition  that  a^n 
shoidd  be  of  bounded  variation  is  that  its  real  and  imaghuiry  ptarts 
should  be  of  bounded  variation. 

This  follows  at  once  from  the  inequalities 

I  ^m       ^m+l  I  ^  j  Cini        (^m+l  \ ;         j  Pm        Pm+l  i  ^  !  ^^ in        ^^9n+l  |> 
I  (^m       C^7n+i  I  ^  I  ^m       ^m+i  \  "i'  [  Pm       Pm+i  \  > 

where  a,,,,  =  «,„  +  i^,„. 

*  Ser.  2,  vol.  1,  pp.  124 — 128.     See  also  'Note  in  addition  to  a  former  paper  on 
conditionally  convergent  multiple  series',  ibid.,  vol,  2,  1904,  pp.  190 — 191. 


of  certain  multiple  series  87 

Theorem  2.  The  necessary  and  sujjicient  cunditiuu  that  a  real 
function  a,„  shoidd  be  of  bounded  variation  is  that  it  should  he  of 
the  form  A^n—AJ,  where  A.^.  and  A  J  are  'positive  and  decrease 
steadily  as  m  increases. 

The  sufficiency  of  the  condition  follows  at  once  from  the 
•inequality 

I  Clm  ~  ftjn+l  !  ^  \A,n,  —  -4,„  +  i)  +  {Ajn   —  A  ,„,+]). 

In  order  to  prove  that  it  is  necessary,  let  us  suppose  that  a,„  is 
of  bounded  variation,  and  let  us  write 

P„i  =  !  flm  -  (I'm+l  I  (an,  "  ««<+!  >  0),      p,„  =  0  (a,,,,  -  a,„+,  <  0), 

Pm'^  !  "m  -  (Im+l  \  (('in  "  «/»+!  <  0),       j)„/ =  0  (a,;,  -  d ,„^.^  >  0), 

B    =  ^'  i)        7?  '  =  S  «  ' 

-'J??;  —  —'  /'n>       -"m        -^  J'n  • 

II!  VI 

Then  B,,,   and  i?„/  are   positive   and  decrease  steadily  as   m   in- 
creases ;   and 

B,n  -  BJ  =  2  ((f„  -  a„,+i)  =  ff,,,  -  a. 

in 

We  may  therefore  take  A,„,=  B„,,  +  G  and  ^j,/=  5,,,' +  C,  where 
C  and   C"  are  suitably  chosen  constants. 

Theorem  3.  //  a,,;  is  of  bounded  variation,  and  Su,,,,  is  con- 
vergent, then  2rt,„M„,,  is  convei^gent. 

Theorem  1  shews  that  it  is  enough  to  prove  this  theorem 
when  a„,  is  real.  Theorem  2  shews  that  it  is  enough  to  prove  it 
when  «„,  is  positive  and  steadily  decreasing.  In  this  form  the 
theorem  is  classical*. 

Lemma  a.  If  2c,„  is  a  divergent  series  of  positive  terms,  we 
can  find  a,  sequence  of  positive  numbers  e,„,  tending  steadily  to  the 
limit  zero,  such  that  2e,„c,„  is  divei'gent. 

Lemma  /3.  If  2c,„,  is  a  divergent  sei'ies  of  positive  terms,  we 
can  find  a  sequence  of  integers  m^  such  that  the  series  Sc,„' ,  where 
Cm'  =  0  if  m  =  nil  and  c,,'  =  c,,,  otherwise,  is  divergent. 

Lemma  a  is  due  to  Abelf.  Lemma  (3  is  quite  trivial,  and  the 
proof  may  be  left  to  the  reader. 

*  See  Bromwieb,  Infinite  Series,  p.  48.  Theorem  3  is  given  by  Dedekind  in  bia 
editions  of  Diiicblet's  Vorlesungcn  iiber  Zahlcntheorie :  see  e.g.  p.  255  of  the  tbird 
edition.  Tbe  central  idea  of  all  sucb  tbeorems  is  of  course  Abel's.  Tbe  line  of 
argument  followed  bere  is  due  substantially  to  Hadamard,  'Deux  theoremes  d'Abel 
sur  la  convergence  des  series'.  Acta  Matheinatica,  vol.  27,  1903,  pp.  177 — 184. 

t  'Sur  les  series',  (Euvves,  vol,  2,  pp.  iy7-~20o. 


88  ilf?'  Hardy,  On  the  convergeyice 

Theorem  4.  If  ^a,f,u,n  ■'*'  convergent  whenever  'lii,,,  is  cuii- 
vergent,  then,  a,„,  is  of  hounded  variation. 

This  theorem  is  due  to  Hadamard*.  We  have  to  shew  that, 
if"  X  I  ^m  —  f'?n+i  j  is  divergent,  v^„  can  be  so  chosen  that  -n,n  is 
convergent  and  1a,nUr,n  is  not.  By  Lemma  a,  we  can  choose  a 
sequence  of  positive  and  steadily  decreasing  numbers  e„,  so  that 
e„,  — *  0  and  Sc,„,  where 

is  divergent.     By  Lemma  /3,  we  can  then  choose  the  sequence  m,- 
so  that  %c„j'  is  divergent.     We  take 

u^  =  U,,     tL,n  =  L\n  -  ?7,„_,  (m  >  1), 

where  Um,.  =  0, 

and  U,n  =  em       

if  m^mi,  the  last  expression  being  interpreted  as  meaning  e„, 
if  a,„  =  a^+i  •     We  have  then 

mi  mi—1  mi-1 

1  1  1 

which  tends  to  infinity  with  i.     Thus  l.amUm  is  not  convergent, 
while  ^Um.  converges  to  zero. 

We  may  call  a,,;,  a  convergence  factor  if  2o„,m,„  is  convergent 
whenever  %Um.  is  so.  Theorems  3  and  4  may  then  be  combined 
concisely  in 

Theorem  5.  The  necessary  and  suffi,cient  condition  that  a^n 
shoidd  he  a  convergence  factor  is  that  it  should  he  of  hounded 
variation. 

Double  series. 

3.  The  convergence  of  a  double  series,  in  Pringsheim's  sense  •]-, 
does  not  necessarily  involve  the  convergence  of  any  of  its  rows  or 
columns  |.  In  this  paper  I  shall  confine  my  attention  to  con- 
vergent series  whose  rows  and  columns  are  convergent  separately  : 
in  this  case  I  shall  say  that  the  series  is  regularly  conve7'gent. 
A  regularly  convergent  double  series  is  also  convergent  when 
summed  by  rows  or  by  columns,  and  its  sum  by  rows  or  by  colunms 
is  equal  to  its  sum  as  a  double  series  §. 

Similarly  I  shall  say  that  a^.^n  tends  regidarly  to  a  limit  if 

lim  a,„,, n  =  «n ,     lim  « „,,, „  =  «,„ , 

*  I.e.  supra.  t  Bromwich,  Infinite  Series,  p.  72. 

+  Bromwich,  ibid.,  p.  74.  §  Bromwich,  ibid.,  p.  75. 


of  certain  multiple  seines  89 

and  the  double  limit 

lim     a^n.,  n=  CL, 

all  exist.     In  this  case  0^  a.nd  «„  tend  to  a  when  m  and  n  tend  to 
infinity. 

Lemma  7.      //  SSum^n  is  regularly  convergent,  to  the  sum  s,  and 

m  n 
1   1 

then,  given  any  positive  number  e,  we  can  find  co  so  that 

I  *m,M       S  I  <  6 

if  either  m  or  n  is  greater  than  co. 

We  may  suppose  s  =  0  without  loss  of  generality.  Since  the 
double  limit  exists,  we  can  choose  w^  so  that  |  6',„,^„  \<  e  if  ni  and  n 
are  both  greater  than  ^i.  When  &)i  is  fixed  we  can  choose  o).,  and 
«»3  so  that  the  inequality  is  satisfied  for  1  ^m  ^(o^,  n>  Wn  and  for 
m  >  &)3,  1  •^  n  ^Wj.  We  can  then  take  to  to  be  the  greatest  of  <wi, 
(On,  and  0)3.  ,  . 

Lemma  S.     In  the  same  circumstances,  we  can  choose  w  so  that 


pq 
m  n 


<  e 


if  p^m,  q^  n,  and  either  m  or  n  is  greater  than  w. 
This  follows  at  once  from  Lemma  7  and  the  identity 
1^        _  , 

m  n 

4.     I  shall  say  that  a^n^n  is  of  bounded  variation  in  (m,  71)  if 

(1)  a^n^n  is,  for  every  fixed  value  of  m  or  n,  of  bounded 
variation  in  ?i  or  m, 

(2)  the  series 

is  convergent.  And  I  shall  say  that  ar,i^n  is  a  convergence  factor  if 
SSa„i_,j,Mm_,i  is  regularly  convergent  whenever  SSw,„,,„  is  regularly 
convergent.  My  main  object  is  to  prove  the  analogue  of  Theorem  5 
for  double  series,  i.e.  to  establish  the  equivalence  of  these  two 
notions. 

VOL.  XIX.   PARTS  II.,  III.  7 


90  Mr  Hardy,  On  the  convergence 

It  will  be  convenient  to  write 

The  condition  that  a^n,n  should  be  of  bounded  variation  is  then 
that  the  series  i  |  A„,  a,„,,,i|,  S|A„a^^,,j|,  and  iS  ]  A„i^,i«,„,„  j 
should  all  be  convergent.  It  is  clear  that  these  conditions  in- 
volve the  regular  convergence  of  a^^n  to  a  limit  a. 

Theorem  6.  If  the  condition  (2)  is  satisfied,  and  a„,,i  and  «!_„ 
are  of  bounded  variation  in  ni  and  n  respectively,  then  a,n,n  is 
of  bounded  variation  in  (m,  n). 

n  —  l 

For  A^a^,„  =  A^a^_i-  1,  A^_^«.^,^, 

v  =  l 
m—\  m— 1  ni-\n-\ 

S  |A^a^,«!^  2  |A^a^,i|+    S    S  |A^_^a^,J, 

/u.  =  l  /u.  =  l  n  =  l  1^  =  1 

so  that  2  I  A^,a^,M  j 

is  convergent. 

Theorem  7.     If  a^n,n  is  of  bounded  variation  in  {m,  n),  then 
a,n=  lim  am.^n,     ««=  lim  cv,n 

are  of  bounded  variation  in  m  and  n  respectively. 


For  a^=  ai  1,—  %  A 


^t=i 


/xfi^Ai, »»' 


W-1  71-1  ao      11-1 

S  I  «;,  —  a^+i|  ^  2  I  A^tti,^  j  +  2    2)  I  Aj^,^a^,„|, 

and  so  2  ]  a^  —  a^+j  j 

is  convergent. 

Theorem  8.  The  necessary  and  sufiicient  condition  that  a,n,n 
should  be  of  bounded  variation  is  that  its  real  and  imaginary  parts 
should  be  of  bounded  variation. 

This  follows  from  Theorem  1  and  the  inequalities 

I  ^m,n(^'m,n  \  ^  j  ^m,  ?i^m,9i  I  "r  |  ^in,n  Hm.,n  |j 
where  am,n  =  '^m„n  +  i^m,n- 


of  certain  multi'ple  series  91 

Theorem  9.  The  necessary  and  sufficient  condition  that  a  real 
function  a.m^n  should  he  of  bounded  va,riation  is  that  it  should  he  of 
the  fo7'm  Am,n  —  A\n,n,  whcrc 

and  A\n^n  satisfies  similar  conditions. 

Suppose  first  that  a-m^n  is  of  tlie  form  indicated.  It  is  plain 
that  the  series 

^^7n-^m,n>       ■^^7i-^7n,nt       ■"-^^m,  ?i-^?n.,  n> 
m  n 

and  the  corresponding  series  formed  from  ^',„„>i,  are  all  convergent. 
Further  we  have 

and  similar  inequalities  for  A„o,„_,i  and  \n,n(^m,n-     Hence  «,„,„  is 
of  bounded  variation. 

Next  suppose  that  A,n,n  is  of  bounded  variation,  and  let 

Pm,n  =  I  ^m,n(^7n,n  \  \^7n,7i^m>7i  ^  ")>       Pm, 7i  ="  "  \^rn, 7i^7n,7i  <  ^)) 
P  7)1,71  ~  I  ^m,7iO-7n,7i  I  ( ^m, Ji C^m, ?i  ?$  ^/i       P  m,7i  ~  ^  \^m,n(^7n,7i  >  ^)- 

Suppose  also  that 

C»    00  00    00 

^-'711,  Jl  —   -^  —  JJlJL,  VI        ^^  771, 71  —  -^  — '  Z'  M,  I'  • 

771  ji  m  rt 

Then  it  is  plain  that 

^7n-Drn,7i^  ^>       ^n^in, 7i-^^i       ^7n,7i-t^7n,n^^^ 

and  that  B'^^n  satisfies  similar  conditions. 
Also 

00    00 

J^7n,  n  —  -O  7JJ ,  jj  =  -^  .i  ^fi,  V  Ojh,  V  ^^  ^m,  ?!.       ^m       ^7i  "r  '^j 
m  w 

(^771,71  ^^  -ttnijTi       -O  7ji,  n  +  ftjn  ~r  ttjj       ft. 

But,  by  Theorems  7  and  2,  we  have 

where  0^.,  CJ,  D,,,  and  D,/  are  positive  and  steadily  decreasing 
functions.     Thus 

^7n,  >i  ^^^  ■^7n,7i       -^  on,  7i  > 

where 

Am,7i  —  Bm^n  "H  ^m  +  Dn  +  Jif,       A  ,n,7i—  B  rn.^n  +  ^m   +  -L'oi   +  -^  > 

£^  and  E'  being  suitably  chosen  constants;  and  it  is  clear  that 
Am,7i  and  A',n,7i  will  satisfy  the  conditions  of  the  theorem  if 
E  and  E  are   sufficiently  large. 

7—2 


92 


Mr  Hardy,  On  the  convergence 


Theorem  10.     //  a.m.,n  *'*  of  bounded  variation,  and  2St/,„,,j,  is 
regularly  convergent,  then  SSa,„,,„M,„,„  is  regidarly  convergent. 

In  virtue  of  Theorem  8,  it  is  enough  to  prove  this  when  o.,n,n 
is  real.     In  virtue  of  Theorem  9,  it  is  enough  to  prove  it  when 

am.,n  >  0,     A„,a,rt,,i  ^  0,     Ana,„,„  >  0,     ^m.,na.m,n  >  0. 

In  the  first  place,  by  Theorem  3,  every  row  and  column  of  the 
series  S2a,„^„iA,«,,i  is  convergent. 

In  the  second  place,  we  have 

m  n  m       n  m  n 


p-l  («.  q  q-l  P  V  pg 

+  2  A^  a^,  g  2  2  %i, j  + ,  2  A^  a^^  ^  22  Uij  +  a^^  ^  2  2  Uij  * . 

m  in  n  ii  m  n  in  n 


It  follows  that,  if  _p  ^  m,  q^n,we  have 


p  q 


<a 


m,n-'^'m,,n> 


2^2^  a^l     y  ^fji,  V 

mn 

where  Hm^n  is  the  upper  bound  of 

1  '^ "         i 
'^'%Ui,j    (/-t  ^  m,  n  ^  v). 

\  mn  I 

Now 

p  q  mn  f    P         1  p     n  m    q^ 

22  a^,  ^iV,"- 22  «M,^ '?''/",>'=     2    2  +  22+22 

1  1        '  11  \in+l  n+1        m+1  1  1  w+l 


II 


and  so 

p  q 

21 
1 1 


1  1        '  '  11 

where  h^^n  is  the  upper  bound  of 

22«t,j 


for  all  values  of  k,  I,  /x,  and  i/  such  that  fi'^k,  v^l,  and  Tom 
or  l>n. 

*  See  pp.  124—125  of  my  paper  quoted  in  §  1,  where  the  general  form  of  this 
identity,  for  multiple  series  of  any  order,  is  given.  Similar  transformations  of 
double  series  were  given  independently  by  M.  Krause,  '  Uber  Mittelwertsatze  im 
Gebiete  der  Doppelsummen  und  Doppelintegrale',  Leipziger  Berichte,  vol.  55,  1903, 
pp.  240 — 263.  See  also  Bromwich,  'Various  extensions  of  Abel's  Lemma',  Proc. 
London  Math.  Soc,  ser.  2,  vol.  6,  1907,  pp.  58—76,  where  further  interesting 
applications  of  the  identity  are  made, 


of  certain  multiple  series    '  93 

Hence,  by  Lemma  8,  we  can  choose  co  so  that 


p  q  m  n 

11  11 


if  m  and  n  are  greater  than  eo.  Thus  the  double  series  is  con- 
vergent, and,  since  its  rows  and  cokimns  are  convergent,  it  is 
regularly  convergent. 

When  a^n,^n  and  its  various  differences  are  positive,  this  theorem 
is  nearly  the  same  as  that  referred  to  in  §  1.  It  is  related  to  the 
latter  theorem,  in  fact,  as  what  Dr  Bromwich  calls  'Abel's  test'  for 
ordinary  convergence  is  related  to  'Dirichlet's  test'.*  The  more 
direct  generalisation  is  as  follows. 

Theorem  11.  If  a,n^n  is  of  bounded  variation  and  tends 
regularly  to  zero,  and 

m  n 

2^  2^  U/j,^  „ 

1  1 

is  bounded,  then  S'Zam,n%n,7i  ^■^  regularly  convergent. 

The  proof  is  similar  to  that  of  Theorem  10,  and  I  need  hardly 
write  it  out  at  length.  The  theorem  shews,  for  example,  that 
the  series 


2S 


cos  (md  +  n^) 
(a  +  mco  +  nw'f ' 


where  6  and  <^  are  real,  w'jui  is  positive  or  complex,  and  the  real 
part  of  s  is  positive,  is  regularly  convergent  except  for  certain 
special  values  of  6,  </>,  and  a;  or  again  that  the  series 

^^       cos  (md  +  n(f)) 


(arn^  +  2bmn  +  cn'-y ' 

*  Theorem  10  itself  does  uot  seem  to  have  been  enunciated  before,  even  in  the 
specialised  form.  The  nearest  theorem  which  I  have  been  able  to  find  is  one 
given  by  G.  N.  Moore,  '  On  convergence  factors  in  double  series  and  the  double 
Fourier's  series',  Tia)is.  Amer.  Math.  >Soc.,  Vol.  14,  1913,  pp.  73 — 101.  Moore's 
theorem  (a  particular  case  of  a  theorem  concerning  Cesaro  summability)  is  as 
follows  :  if 

(1)  23«„j,  ,j  is  convergent  as  a  double  series  in  Priugsheim's  sense, 

VI  n 

(2)  |2S«^^J<:7i, 

(3)  ^m,  »  ^  0, 

00  00 

(4)  lim       2    |«„,,,J  =0,   lim       S|a„^,„|=0, 
m-^cc  »=1  w-^-a)m  =  l 

(5)  22  I  M^,  „  a,„,  „  I 
is  convergent,  then  22a„j  „  Um,n  *®  convergent. 


I 


94  Mr  Hardy,  On  the  convergence  H 

11 
where  6,  (f),  a,  h,  and  c  are  real,  and  a,  ac  -  ¥,  and  the  real  part  of  s 
are   positive,   is  regularly   convergent    except  for  certain  special 
values  of  9  and  0.     In  either  of  these  series,  of  course,  the  cosine  \ 
may  be  replaced  by  a  sine. 

In  order  to  prove  the  converse  of  Theorem  10  we  require  two 
lemmas  analogous  to  Lemmas  a  and  yS. 

Lemma  e.  If  SSc^.n  is  a  divergent  series  of  positive  terms, 
we  catt  find  €„i^n  so  that  (1)  e^^^n  decreases  luhen  ni  or  n  increases, 
(2)  €m,n  tends  regularly  to  zero,  and  (3)  the  series  Sl,e^n,nGm,n  is 
divergent. 

(1)  Suppose  first  that  at  least  one  row  or  column  of  the 
original  series,  say  the  vth  row  2c„i,  ^,  is  divergent.  By  Lemma  a, 
we  can  choose  a  steadily  decreasing  sequence  97^,  with  limit  zero, 
so  that  XvmCm,p  is  divergent.     We  take 

€vi,  n  =  Vm  (n  ^  v),       €,n,  n  =  0  {u  >  v), 

and  it  is  plain  that  the  conditions  of  the  lemma  are  satisfied 

(2)  Suppose  that  every  row  and  column  is  convergent; 
and  let 

(m)  (w) 

Then  'S,j„^  is  divergent.  We  choose  a  steadily  decreasing  sequence 
r]m  so  that  'S^Tjmjvi  is  divergent.     Then  SSc'„,,,„,  where 

is  divergent;  and  so  Xjn,  where 

yn  ^  ^  Vm  Cm,  n  j 
(m) 

is  divergent.  We  now  choose  a  steadily  decreasing  sequence  ^„, 
with  limit  zero,  so  that  ^^nln  is  divergent.  It  is  clear  that,  if 
we  write 

//  y  

(^  m,n  —  Vmbn^^m.jn  —  ^?w, n C^/i,, ^ , 

all  the  conditions  of  the  lemma  will  be  satisfied. 

Lemma  ^.     If  SSc^.w  is  a  divergent  series  of  positive  terms,  we 
can  choose  a  sequence  of  pairs  of  integers  (nii,  Ui),  tending  to  infinity  ! 
luith  i,  so   that  the  series  X%c\n,n,  where  c'm,n  =  0  if  771  =  nit,  n^ni 
or  m^nii,  n  =  ni,  and  c,n,n  =  c,,,,,,,,  otherwise,  is  divergent. 

The  modification  to  be  made  in  the  series  is  effected  by 
drawing  perpendiculars  on  to  the  axes  from  the  points  {mi,  n^), 
and  annulling  all  terms  which  correspond  to  points  on  these 
perpendiculars.     Let  a^  denote  the   sum   of  the    terms   whose 


of  certain  multiple  series  95 

representative  points  lie  on  the  perpendiculars  from  (m,  m)  on 
to  the  axes.  Then  So"„i  is  divergent.  Applying  Lemma  0  to  this 
series  we  obtain  the  construction  required,  nii  being  in  fact 
always  equal  to  Ui. 

Theorem  12.  If  S2a,„.^,i,i<„,,,j,  is  regularly  convergent  ivhen- 
ever  SSif.„i.,,i  is  regularly  convergent,  then  a,n,n  "^^  ^f  hounded 
variation. 

In  the  first  place  it  follows  from  Theorem  4  that  ■u,„,^n  is,  for 
every  value  of  n  (or  m),  of  bounded  variation  in  m  (or  n).  It 
remains  only  to  shew  that  SS  |  A,„^„a,„,_,i  |  is  convergent. 

Suppose,  on  the  contrary,  that  it  is  divergent.  By  Lemma  e, 
we  can  choose  a  sequence  of  positive  numbers  €,„,„,  tending 
regularly  to  zero,  so  that  2Sc,rt,n,  where 

is  divergent.  We  can  then  modify  this  series  as  in  Lemma  ^ 
without  destroying  its  divergence. 

Now  let 

m  n 
U        —^l.u 
1    1 

and  suppose  that 

if  m  =  mi,  n  ^  ni  or  m  ^  mi,  n  =  ni,  and  that  otherwise 


^  in   n,   — 


this  last  formula  being  interpreted  as  meaning  e^yi^n  if 

These  equations  define  u,n,^n  uniquely  for  all  values  of  m  and  n, 
and  it  is  plain  that  U'm,n  tends  regularly  to  zero,  so  that  2S^/,„,,t  is 
regularly  convergent.     On  the  other  hand 

^  -^ '■hn,n  ^'■m,n  "i        ■^    '^m,n  ^i')n,n  ^ m,n —       -^       ■^    ^  w,n> 

11  11  11 

which  tends  to  infinity  with  i.     Thus  ^'%a„r,nUm,n  is  not  convergent. 
This  proves  Theorem  12.     Combining  it  with  Theorem  10  we 
obtain  the  analogue  of  Theorem  5,  viz. 

Theorem  13.  The  necessary  and  sufficient  condition  that  am^n 
should  be  a  convergence  factor  is  tJiat  it  should  be  of  bounded 
variation. 


96  Mr  Watson,  Bessel  functions  of  large  order 


* 


Bessel  functions  of  large  order.  By  G.  N.  Watson,  M.A., 
Trinity  College. 

\^Received  14  June  1917.] 

1.  When  the  order  of  a  Bessel  function  is  large,  the  asymp- 
totic expansion  of  the  function  assumes  various  forms  depending 
on  the  values  of  the  ratio  of  the  argument  to  the  order  of  the 
function.  The  dominant  terms  of  the  asymptotic  expansions  are 
given  by  the  formulae  : 

(i)     When  n  is  large,  x  is  fixed  and  0 ^x<  1,  then 
Jn  {nx)  ~  (27rw)"*  (1  -  a^y^x""  [1  +  V(l  -  a?)]"'  exp  [n  V(l  -  x% 
(ii)     When  n  is  large,  x  is  fixed  and  x>l,  then 

Jn{nx)<^{^'jrny^{x'^-  l)~^cos  [?i  V(*'^  -  l)-n^e(r^x-  lir]. 

_  2. 
(iii)     When  n  is  large  and  €  =  0{n   ^),  then 

J^{n  +  ne)^T(i)/{7r2is^n^]. 

The  corresponding  complete  asymptotic  expansions,  valid  for 
general  complex  values  of  n  and  x,  have  been  given  by  Debye*. 
Accounts  of  the  history  of  the  approximate  formulae  are  to  be 
found  in  Debye's  memoirs  and  also  in  two  papers  f  which  I  have 
published  recently. 

It  is  evident  that  there  are  transition  stages  between  the 
domains  of  validity  of  the  three  formulae  quoted ;  and  not  much 
is  known  about  the  behaviour  of  J„  (jix)  in  these  transition  stages. 
Consequently  I  propose  to  establish  approximate  formulae  (involv- 
ing Bessel  functions  of  orders']  +  ^)  which  exhibit  the  behaviour 
of  the  Bessel  function  right  through  the  transition  stages.  These 
formulae  are  more  exact  forms  of  some  approximations  which 
Nicholson  §  obtained  some  years  ago  without  estimating  the 
margin  of  error  or  the  precise  ranges  in  which  the  results  were 
valid. 

*  Math.  Ann.,  lxvii.  (1909),  pp.  535—558.   Milnchen.  Sitzungsberichte  [5],  1910. 

t  Proceedings,  xis.  (1916),  pp.  42—48.  Proc.  London  Math.  Soc.  (2),  xvi.  (1917), 
pp.  150—174. 

J  These  functions  have  been  tabulated  by  Dinnik,  Archiv  der  Math,  und  Phys., 
XVIII.  (1911),  p.  337. 

§  Phil.  Mag.,  Feb.  1910,  pp.  228—249. 


Mr  Watson,  Bessel  functions  of  large  order  97 

The  approximations  which  I  shall  obtain  are  derived  by 
shewing  that  certain  integrals  of  Airy's  type*  are  effective 
approximations  to  the  integrals  which  occur  in  Debye's  analysis. 
It  will  be  assumed  that  the  reader  is  familiar  with  Debye's 
memoirs,  although  it  seems  desirable  to  modify  the  notation  to 
a  considerable  extent.  The  formula  for  Jn  (nx),  when  x^l,  is 
of  importance  in  connexion  with  the  maxima  of  the  Bessel 
function  f. 

The  two  formulae  which  will  be  obtained  in  this  paper  are  as 
follows : 

(I)  When  a  ^  0, 

Jn  ( n  sech  a)  ~  27r~^  3  "  ^  tanh  a 

X  exp  [n  (tanh  ol  +  ^  tanlr'  a  —  a)} .  if  ^  {^n  tanh*  a), 

where  the  error  is  less  than  37i"^  exp  [?i(tanh  a  —  a)|,  and  K,,,{2) 
denotes  the  Bessel  function  of  Basset's  type  (see  §  6). 

(II)  When  0  ^ /3  ^  \7r, 

Jn  {>i  sec  /3)  ~  ^  tan  /3  cos  [n  (tan  ^  —  ^  tan^  yS  —  /3)} 
X  [J_  I  (-3-*^  tan=^  ^)  +  /,  (i/i  tan*  /3)] 

+  3"  ^  tan  yS  sin  [n  (tan  ^-  ^  tan*  j3  -  /3)} 
X  [/_ ,  (iw  tan*  /S)  -  J^  O  tan*  y8)], 

where  the  error  is  less  than  24/?i. 

Part  I.     The  value  of  Jn(n.v)  ivhen  O^^-^l. 
2.     We  take  Sommerfeld's  integral 


The  stationary  points  of  x  sinh  w  —  w,  qua  function  of  w,  are 
given  by  cosh  w  —  l/x ;  accordingly  we  replace  x  by  sech  a,  where 
a  ^  0  ;  and  then,  putting  w^ot  +  t,  we  have 

jiTTl  J  00  -ni 

+  n  (sinh  t  —  t)]  dt. 

The  exponent  has  a  stationary  point  at  ^  =  0,  and  the  method 
of  steepest  descents  provides  us  with  the  contour  whose  equation  is 

/  {tanh  a  (cosh  t-\)-\-  (sinh  t-t)]  =  0. 

*  These  integrals  have  been  expressed  in  terms  of  Bessel  functions  by  Nicbolsou, 
Phil.  Mag.,  July  1909,  pp.  6—17,  and  by  Hardy,  Quarterly  Journal,  xli.  (1910), 
pp.  226—240. 

+  Proc.  London  Math.  Soc.  (2),  xvi.  (1917),  p.  169. 


98  Mr'   Watson,  Bessel  functions  of  lar-ge  oi'der 

The  portion  of  this  curve  which  is  suitable  for  our  purposes 
consists  of  an  arc*  on  the  right  of  the  imaginary  axis  in  the  ^-plane 
with  its  vertex  at  the  origin  and  with  the  lines  I  {t)=  ±tt  as 
asymptotes. 

If  we  write  t  =  u  +  iv,  where  tt,  v  are  real,  the  equation  of  the 
curve  becomes 

cosh  (a  +  u)  =  V  cosec  v  cosh  a. 

We  shall  put 

tanh  a  (cosh  t  -  1)  +  (sihh  t  —  t)  =  —  r, 

so  that  as  t  traverses  the  contour  t  diminishes  from  +  x  to  0,  and 
then  increases  to  +  oo  ;  and  therefore 

Jn  (^'^)  =  o-^  e«(t^'ii^*-«)  I  j      +       \  e""^  (dt/dr)  dr ; 

in  the  first  integral  v  ^0  and  in  the  second  integral  v  ^  0. 
Now  define  T  by  the  equation  j- 

1^2  tanh  a  +  ^T'  =  -T. 

A  contour  in  the  T-plane  on  which  t  is  real  is  a  semi-hyperbola 
touching  the  imaginary  axis  at  the  origin  and  going  off  to  infinity 
in  directions  inclined  ±  ^tt  to  the  real  axis.     If  we  write 

T=U+iV, 

where  U,  V  are  real,  the  equation  of  the  hyperbola  becomes 

Utanha  +  ^U''  =  iV-\ 

Taking  the  semi-hyperbola  as  the  T-contour,  we  shall  shew 
that  an  approximation  to 

foD+iri  /•  00  exp  ( Jtt?') 

e-'^-'dt  is  e-''^dT. 

J  CO  -  Tii  ■'  00  exp  ( -  JttO 

It  is  easy  to  see  that  the  difference  of  these  integrals  is 
D  idt     dT)  , 


{] ^      Jo  ]  (dr      dr] 


and  so  the  problem  before  us  is  reduced  to  the  determination  of 
an  upper  bound  for  \d{t—  T)/dT  |. 

*  This  curve  is  derived  from  the  curve  shewn  in  fig.  4  (p.  541)  of  Debye's  first! 
paper  by  turning  it  through  a  right  angle  and  talcing  the  origin  at  the  vertex.  The! 
degenerate  case  when  a  is  zero  is  shewn  in  fig.  5.  \ 

t  Since  T  =  ^t^  ta,nh  a +  lt^  +  0  (t*)  when  |  1 1  is  small,  the  curve  in  the  T-plaue 
closely  resembles  the  curve  in  the  i-plane  near  the  origin ;  and,  the  parts  of  the 
curves  near  the  origin  being  the  most  important  when  n  is  large,  we  are  obviously 
able  to  anticipate  that  the  integrals  under  consideration  are  approximately  equal. 


Mr   Watson,  Bessel  functions  of  large  order  99 

3.  We  shall  now  shew  that,  whenever  t  ^  0  and  when,  corre- 
sponding t(^  any  given  value  of  t,  we  choose  V  to  have  the  same 
sign  as  v,  we  have  the  inequality 

\d{t-  T)}dT  I  ^  Stt. 

Since,  corresponding  to  any  value  of  t,  the  two  values  of  t  are 
conjugate  complex  numbers  (and  similarly  for  T),  it  is  evidently 
sufficient  to  prove  this  inequality  when  v  and  V  are  both  positive. 

On  comparing  the  values  for  r  in  terms  of  t  and  T,  we  perceive 
that 

'  ( 7'  -  0  {H^  +  0  tanh  a  +  1  (T'  +  Tt  +  f')} 


tanh  a  (cosh  ^  -  1  -  ^t-)  +  (sinh  t-t-  ^t-'). 


Also 
d(t-  T)jdT  =  { jTtanh  a  +  ^T''}~'  -  {sinh  t  tanh  a  +  (cosh  t  -  1)}-^ 
^  t-T 

T  {sinh  t  tanh  a  +  (cosh  ^  —  1 )] 

^^(^-r)  +  (sinh  t  -  i)  tanh  ajl^(cosh  t-l-^f) 
^    TXta'nh  a  +  iT)  {sinh  « tanli^+ (coshT-nf)}      " 
Now 

I  sinh  t  tanh  a  +  (cosh  t—l)\ 

—  sech  a  v/[(cosh  u  —  cos  v)  {cosh  (2a  +  w)  —  cos  vW ; 
and  since 

{cosh  (2a  +  (t)  —  cos  v]  —  cosh-  a  (cosh  ii  —  cos  v) 

=  sinh-  a  (cosh  m  -t-  cos  y)  +  sinh  2a  sinh  a 

and 

{cosh  (2a  +  u)  —  cos  v]  —  sinh-  a  (cosh  a  +  cos  y) 

=  cosh^  a  (cosh  <(  —  cos  v)  +  sinh  2a  sinh  a 

we  6'ee  that  \  sinh  ^  tanh  a  +  (cosh  ^  —  1)  |  exceeds  both 

cosh  ^^  —  cos  V  =  I  cosh  i  —  1 1 

and  also         tanh  a  \/(cosh-  u  —  cos-  y)  =  tanh  a  |  sinh  ^  j . 

We  now  divide  the  range  of  integration  into  two  parts,  namely 
T  ^  1  and  0  :^  T  ^  1. 

4.     Consider  first  what  happens  when  t^  1. 
If  I  T|  ^  1,  we  have  (on  the  T'-contour) 

T  =  I  IT-  tanh  a  +  1^=*  |  <  ^  +  i  <  1. 


100  Mr   Watson,  Bessel  functions  of  large  order 

Also,  if  I  ^  I  ^  1,  we  have  (on  the  ^-contour) 

T  =  I  (cosh  ^  -  1)  tanh  a  +  (sinh  t-t)\^  S   \t  i '"/''* !  ^  e  -  2  <  1. 

Hence,  when  r^l,  we  must  have  both  \T\^1  and  also  \t\^l. 
But,  when  |  T|  ^  1,  since  U^O,  we  have 

I  (dr/dT)  I  =  I  Ttanh  a  +  ^-T- 1  ^  ]  tanh  a  +  ^T\>^. 

Also,  when  |  i  |  ^  1,  we  have  u  or  v  (or  both)  greater  than  l/\/2, 
and  we  always  have  v  less  than  vr. 
Hence,  by  the  result  of  §  3, 

I  (dr/dt)  I  =  I  sinh  t  tanh  a  +  (cosh  t  —  1)\^2  (sinh'^  ^u  +  sin^  ^y), 

and  this  exceeds  the  smaller  of 

2sinhni/V8),     2sin^(l/v'S). 
Consequently 

i  (dt/dr)  I  ^  1  cosec^  (l/\/8)  =  4-14  <  27r  -  2. 
Therefore,  when  t  ^  1,  we  have  \d(t  —  T)/dT  \  <  ^ir. 
We  shall  make  use  of  this  inequality  in  §  6. 
5.     Consider  next  what  happens  when  0  ^  t  ^  1. 
If  I  T|  ^  2,  we  have  (on  the  I^-contour) 
T  =  I  i-r^  tanh  a  +  iTM  ^  4  !  1  tanh  a  +  ^T\>  l\  Tl^^. 

Also  noting  that  u  and  v  increase  together,  when  v  "^  ^tt,  we 
have  (on  the  ^-contour) 

T  =  u  +  tanh  a  —  cos  v  sech  a  sinh  (a  +  it) 
^  M  +  tanh  a 
^  tanh  G  —  a  +  log  {-^-TT  cosh  a  +  \/(l7r"  cosh"  a  —  1)}, 

on  expressing  u  in  terms  of  v  and  noting  that  v  cosec  v  exceeds  -^tt. 
This  function  of  a  increases  with  a  and  so  it  exceeds 

log{j7r  +  V(i7r^-l)}  >1- 
Hence,  tuhen  r  ^  1,  we  ??iwsi  have  both  \T\^2  and  also  v^^tt. 
Further,  when  v^I-tt,  we  have 

cosh  u  ^  sech  a  cosh  (a  +  it)  =  v  cosec  w  ^  I-tt  <  cosh  I'l, 
so  that  \t\"  -^  ^TT'  -h  (I'ly  <  4,  a7id  therefore  \t\<2. 

That  is  to  say,  when  r^  1,  neither  \  T\  nor  j  t  j  exceeds  2. 

Also,  for  all  values  of  t, 

du/dv  =  {1  —V  cot  v)/i\/(v"  —  sin^  v  sech^  a) 
^  (1  —  ?;  cot «;)/«;  ^  ^-v, 
and  so  u'^^v'^. 


Mr    Watson,  Bessel  functions  of  large  order  101 

Further,  when  v  ^  ^tt,  we  have*  1  —  v  cot  v  ^  ^J{v-  —  sin-  v),  and 
so  dujdv  ^  1 ,  i.e.  v  ^  w,  whence  at  once  we  have  v  \/2  '^\t\. 
Next,  when  |  ^  |  $;  2,  we  have 

\^m\xt\^\t\[l  -  ^\t\^  -  -,^^,\t\*  -  ...] 

In  like  manner  we  may  prove  that,  when  |  ^  |  ^  2, 

I  cosh  i  -  1  I  ^  4  I  ^  J-/18  ^l\t\\     I  cosh  t-\-\t'\^\t  IV20, 
i  sinh  t-t\^^\t  IV24,     |  sinh  t-t-^,f\^\t  I7IO8. 

We  are  now  in  a  position  to  obtain  an  upper  bound  for  |  7^-  ^  |. 
It  is  first  evident  that 

\\{T+t)  tanh  a  +  1  (^'  +  ^«  +  t')  I 

^  /  {i  ( 2^  +  0  tanh  «  +  1  ( r^  +  r^  +  01 
^  I V  tanh  OL-^  ^uv 
^  |-w(tanha  + i-y-) 

^|i|(tanha  +  yVl^l')/\/8 
^U|2(itanha  +  T'H|^I)/V8. 
Hence,  by  the  result  stated  in  §  3, 

\l-t\^d>    \t\    -ytanha+ 1^1/18     ^^^^^l^l    ^i|«l. 

To  obtain  a  stronger  inequality,  we  write  the  equation  of  §  8 
in  the  modified  form 

{T-t){T  +  t)  {1  tanh  a  +  ^{T-\-t)] 

=  -{T-  tfl24>  +  tanh  a  (cosh  t-l-^f-)  +  (sinh  t-t-  ^t^). 

The  expression  on  the  right  does  not  numerically  exceed 

I  ^  17192  +  tanh  a.\t  \'/20  +  \  t  IVIOS  ^  tanh  a .  1 1\'/20  +  |  ^1748  ; 

and  since  \{T  +  t)  [^tixnh.a  + ^(T  +  t)]\  exceeds  both  -H^ltanha 
and  also  ^|^1",  we  see  that 

\T-t\^{j^^+i)\t\'=i\tm5. 

If  we  now  f mother  restrict  t  so  that  |  ^  |  ^  1,  the  last  inequality- 
gives 

\{-(T-  i)V24  +  tanh  a  (cosh  t-1-  ^f-)  +  (sinh  t-t-  |^«)}  | 

^  4-^ I « 1 7(24  .  150  +  tanh  a.\t  |720  +  |  ^  J7108 
^tanha.  1^1720 +  1^17104, 

*  Since    -j~  {siii'^j'  (1  -  v  cot  v)^  -  sin-i;  {v-  -  sin-  c)  j-  =  --  2  sin  2v  .  (v^  -  sin^ u)  <  0 
when  0  <  u  <  Itt. 


102  Mr   Watson,  Bessel  functions  of  large  order 

and   this  inequality,  combined    with    the    modified   form   of  the 
equation  of  §  3,  gives 

I  T- ^1^(1/10  + 1/13)  jip'^i^lVS. 


Hence,  when  |^!^1,  \T-t\^^\t\,  and  so  \T\^^\t\;  and 
therefore,  when  U  |  ^  1,  \T\  exceeds  the  value  which  it  has  when 
1^1  =  1,  and  so,  a  fortiori,  |  T\  ^  |. 

It  now  follows  that,  when  both  r  and  \t\  do  not  exceed  1, 
we  have 


d^_dT  ^ i  1 1 

dr      dr 


II  ^  I .  I  (sinh  t  tanh  a  +  (cosh  i  —  1)}  | 

1 1  [yiO  +  5  tanh  a  \  t  |V24  +  1 1  |V20 
"^  (8  1^  IV25)  I  {sinh  t  tanh  a  +  (cosh  t-l)]\ 
=  (23  i  ^  1732  +  125  tanh  a  1 1 1/192} 
-^  I  {sinh  t  tanh  a  +  (cosh  t—  1)}  j . 

Since  the  denominator  exceeds  both  1 1  ^  |  lanh  a  and  i  |  ^  |",  we 
see  that 

\d{t-  Tydr  I  ^  (23/8)  +  (125/128)  <  27r. 

If  T  ^  1  and  1  ^  j  ^  I  ^  2  we  use  the  second  expression  of  §  3  for 
d{t—  T)ldr.  Replacing  \T—t\  in  the  numerator  by  4  |  ^1^15 
and  \T\  in  the  denominator  by  |,  we  get  in  a  similar  manner 

\d{t-  T)ldr  \^[\t  IV3  +  125  tanh  a .  |  i  |7192  +  11 1  ^  |  V60| 
-^  j  {sinh  t  tanh  a  +  (cosh  t  —  1 )}  | 
^  4  {I  ^  i/3  +  1 1  I  ^  I V60}/13  +  125  i « I-/128 
<37r. 

6.  It  is  obvious,  from  the  results  of  §§  4,  5,  that,  luhenever 
T  ^  0,  we  have 

\d{t-  T)ldT  I  <  Stt  ; 

and  from  this  result  we  have 


27r 


0       r]     „   idt     dT\  ,1     .,  /•"     „    ,       ^, 


The  evaluation  of  I  e""**^(ZT  presents  no  special  points 

J  CO  exp(  — -J-irt) 

of  interest ;  the  simplest  procedure  is  to  modify  the  contour  into 
two  rays,  starting  from  th^  point  at  which  T  =  —  tanh  a  and 
making  angles  +  -J^tt  with  the  real  axis. 


Mr   Watson,  Bessel  functions  of  large  order  103 

If  we  write  T  =  —  i&xih.  a  ■\-  ^e*>'  on  the  respective  rays,  the 
integral  becomes 

/•OO 

e'-'^*'  exp  (i?i  tanh-'  a)  A     exp  { -  i?ip  -  ^n^e^""'  tanh^  a]  d^ 

JO 

-  g-i'^'  exp  (1??,  tanh'  a) .        exp  { -  i?^p  -  i^j^^-i'^''  tanh-  aj  d^. 

These  are  integrals  of  Airy's  type  ;  on  expanding 

exp  (—  |n^e*  ■•'^*  tanh-  a) 

in  powers  of  tanh  a  and  integrating  term-by-term — a  procedure 
which  is  easily  justified — we  get  on  reduction 

Itti  tanh  a .  exp  (i?i  tanh^  a  ) .  [/_  j^  (^n  tanh'  a)  —  /j^  (J-?i  tanh"  a)], 

where,  in  accordance  with  the  ordinary  notation, 

On  introducing  Basset's  function  K^Az),  defined  as 
^ TT  cot  niTT  [/_^  (z)  -  /,„  (z)], 

we  obtain  the  final  formula 

2 
t/^  (n  sech  a)  = ~  [tanh  a  exp  {??  (tanh  a  +  i^  tanh'  a  -  a)] 

TT  V  "  '^ 

X  Kx  Qn  tanh'  a)]  -|-  SdiU-^  exp  {w  (tanh  a  —  a)], 
where  |  ^i  j  <  1. 

When  w  is  large  the  ratio  of  the  error  term  to  the  dominant 

term  is  of  order  n~^\/tanha,  n~^,  vT^,  according  as  n  tanh'' a  is 
large,  finite  or  small. 

The  formulae  (i)  and  (iii)  of  §  1  agree  with  this  result  when  a 
is  finite  and  when  n  tanh'  a  is  small,  respectively. 

Part  II.     The  value  of  Jn(nx)  when  x^l. 

7.  It  is  convenient  to  regard  Hankel's  solutions  of  Bessel's 
equation,  iT^'^'  and  Hn^^^  as  fundamental.  The  ordinary  solutions 
are  expressed  in  terms  of  these  functions  by  the  equations 

Jn  (nx)  =  i  {^,,w  (nx)  +  HJ'^  {nw)], 
The  integral  formulae  of  Sommerfeld's  type  are 

1        roo+ni 

Hn^'^  (nx)  =  -—.  e«  (xsinhw-tv)  dyj^ 

TT'l  J  _oo 

2      roo—iri 


104  Mr   Watson,  Bessel  functions  of  large  order 

The  stationary  points  of  xi^m\iw  —  iv,  qua  function  of  w,  are 
given  by  cosh  ?<;  =  !/«.  As  0  <\lx^l,  we  put  x  =  sec/3  where 
0 ^ /3 <  I^TT ;  and  two  stationary  points  are  given  by  w=  ±  ^i. 

Now  it  has  been  shewn  by  Debye  that  a  branch  of  the  curve  * 
I{x  sinh  lu  -  w)  —  I  {x  sinh  ijS  —  i/3) 

is  a  suitable  contour  for  HJ^\  and  the  reflexion  of  this  contour  in 
the  real  axisf  is  a  suitable  contour  for  HJ-K 

On  making  a  change  of  variable  by  writing  w  =  t  +  i^,  we 
have 

1  rQO+7r?-(/3 

where  i  tan  /3  (cosh  t  —  1)  +  sinh  t  —  t  =  —  T. 

If  we  put  t  =  'u  +  iv,  where  u,  v  are  real,  the  equation  of  the 
contour  is 

cosh  w  =  (sin  ^  +  v  cos  /3)  cosec  (v  +  /3), 

and,  on  the  contour, 

r  =  u  —  sec  /S  sinh  u  cos  (v  +  /3). 

When  V  is  given,  cosh  u  is  given  and  the  sign  of  u  is  ambiguous  ; 
we  take  u  to  have  the  same  sign  as  v,  in  order  that  the  contour  may 
be  of  the  requisite  type. 

Next  define  T  by  the  equation 

^TH  tan  /3  +  ^T' =  -  r. 

We  write  T^U+iV,  where  U,  V  are  real;  a  contour  in  the 
T- plane  on  which  t  is  positive  is  that  branch  of  the  cubic  :|:,  whose 
equation  is 

(t/^  _  T/2)  tan  ^  +  1  F(3[7-  -  V)  =  0, 
which  passes  from  —  cc  —  i  tan  ^  through  the  origin  to  x  exp  (^Tri). 

Taking  this  curve  as  the  contour,  we  shall  shew  that  an 
approximation  to 

r<x+Tri—ip  rco  exp  (Jn-i) 

e-''^dt  is  e-'^^dT. 

.'-cxj-i/S  J  -00— i tan/3 

*  This  curve  is  derived  from  the  curve  shewu  in  fig.  2  (p.  540)  of  Debye's  first 
paper  by  turning  it  through  a  right  angle  and  taking  the  origin  at  tlie  node.  The 
reader  will  observe  that  tlie  character  of  the  contour  has  changed  with  the  passage 
of  X  through  the  value  unity. 

t  Since  iJ^*^),  HJ")  are  conjugate  complex  numbers  when  n  and  x  are  real,  it 
will  be  sufficient  to  confine  our  attention  to  HJ^^. 

X  Of  course  t  is  real  on  the  whole  cubic ;  as  T  traverses  the  specified  portion  of 
it,  T  decreases  from  +  oo  to  0  and  then  increases  to  +  oo  . 


Mr   Watson,  Bessel  functions  of  large  order  105 

8.     Before  proceeding  further,  we  shall  shew  that  the  slopes  of 
the  contours  in  the  t-plane  and  in  the  T-plane  never  *  exceed  s/S. 
If  we  write 

(sin  ^  +  V  cos  /3)  cosec  (v  +  /3)^  ^|r  (v), 

we  have  dv  ^sinhu^  ^  [{^(^)}2  _  i]l  ^ 

du       '\Jr' (v)       ~  ■\lr'(v) 

Now 

yjr'  (v)  =  cosec  (/3  +  v)  {cos  yS  —  cot  (^  +  v)  (sin  l3  +  v  cos  /8)}, 

and  so  -yjr'  (v)  is  positive  when  /3  +  v  is  an  obtuse  angle.  When 
0  ^/3  +  v^^TT,  however,  we  find  that 

cos  /3  tan  (/3  +  v)  —  (sin  13  -\-  v  cos  /3) 

is  an  increasing  function  which  vanishes  with  v.  Hence  yjr' (v) 
has  the  same  sign  as  v  (and  therefore  the  same  sign  as  u),  and 
consequently 

dv  ^[{ylr(v)\'-lf 
du  I  yjr'  (v)  I 

It  is  therefore  necessary  to  prove  that 

i.e.  that  x  (^)  = '^  li"'  W'  "  if  (^)}'  +1^0. 

Now  %  (0)  =  0,  and  it  is  consequently  sufficient  to  shew  that 
X  {^)  h'^s  the  same  sign  as  v.     Since 

X{v)  =  2y{r\v){Sf"(v)-ylr{v)} 

and  yfr' {v)  has  the  same  sign  as  v,  it  is  sufficient  to  prove  that 

Srfr"  (v)  -  f  (v)  ^  0. 

Since  yjr  (v)  sin  (v  +  /8)  reduces  to  a  linear  function  of  v,  its 
second  derivate  vanishes,  and  so  the  inequality  to  be  proved 
reduces  to 

f  {v)  -  3-f'  (v)  cot  (v  +  yS)  ^  0, 
i.e.  to 

(sin  ^+v  cos  /3)  {1  +  3  cof^  {v  +  /3)}  -  3  cos  ^  cot  (v  +  /3)^0. 

But 

sin  yS  +  V  cos  yS  -  3  cos  ^  cot  {v  +  /3)/{l  +  3  cot^  {v  +  /3)} 

has  the  positive  derivate  4cos  yS  (1  +  3  cot"(w  + /3)}~^  and  is 
positive  when  v  =  —  ^;  hence  it  is  positive  throughout  the  range 
—  ^^v-^TT-^.     And  this  is  the  result  which  had  to  be  proved. 

*  In  the  limiting  case  when  ^  =  0,  the  ^-contour  has  slope  ^3  immediately  on 
the  right  of  theorigin,  and  the  T-contour  consists  of  the  rays  arg  r  =  0,  arg  T  — ^tt  ; 
so  there  is  no  better  inequality  of  the  form  stated. 

VOL.   XIX.   PARTS  II.,  III.  8 


106  Mr   Watson,  Bessel  functions  of  large  order 

In  like  manner,  we  find  that 

dV  _  (tan;e  +  F)^(tan/8  +  ^7)^ 

^dU~     tan'/S  +  Ftan/S  +  iV-     ' 

and  it  may  be  proved  by  quite  simple  algebra  that  the  square  of 
this  last  fraction  does  not  exceed  3. 

From  the  results  just  proved  it  follows  on  integration  that 

\v\^\u\^/S,     |F|^|i7!V3, 
and  hence 

h'!^iUI>     \v\^^\t\^3,     \U\^^\T\,     |Fi$i|TlV3. 

9.  We  now  return  to  the  integrals  of  §  7.  As  in  the  corre- 
sponding work  of  §§  2 — 3,  we  have  to  obtain  an  upper  bound  for 
\d{T  —  t)ldr  i ;  we  shall  in  fact  shew  that  this  function  does  not 
exceed  127r. 

We  notice  that  formulae  corresponding  to  those  given  in  |  3 
are 

{T-t){hAT+t)i\>?.u^-^^{T^^Tt  +  1?)] 

=  i  tan  yQ  (cosh  t-l-hP)  +  sinh  t-t-  ^t\ 
d{t-  T)/dT 

=  {iT  tan  /3  +  ^T']-^  -  {{  sinh  nan  /3  +  (cosh  t  -  1  )]-^ 

t-T 

"  T  [i  sinh  t  tan  ^  +  (cosh  ^  -  1)} 

^t{t-T)  +  i  (sinh  t  -  t)  tan  ^  +_(cosliJ^-  1  -  ^i') 
"^     T {%  tan  y8  +  ^ T)  (TsmhTtan  yS  +  (coshl  -1)} 

Now 

I  i  sinh  t  tan  /3  +  cosh  t  —  1\ 

=  sec  13  V[(cosh  u  —  cos  v)  {cosh  v  —  cos  (2/3  +  w)|], 
and  since 

(cosh  u  —  cos  (2/3  +  v)]  —  cos-  B  (cosh  ii  —  cos  v) 

=  sin^  /3  (cosh  u  +  cos  v)  +  sin  2/3  sin  v 
^  (1  +  cos  v)  {sin-  /S  +  sin  2/3  tan  -^  vj 
^  (1  +  cos  w)  {sin- 13  -  sin  2y3  tan  ^^] 

we  have 

.  I  i  sinh  t  tan  /3  +  (cosh  ^  —  1)  |  ^  cosh  w  —  cos  «  =  |  cosh  t  —  l\. 
Also 

cosh  w  -  cos  (2/3  +  ?;)  ^  2  sin=  (^8  +  -iwi  ^  2  sin- 1/3, 


Mr  Watson,  Bessel  functions  of  lar^ge  order  107 

and  so 

I  i  sinh  t  tan  /3  +  (cosh  t  —  l)\^sm^^  sec  /S  V[2  (cosh  it  —  cos  v)} 

^  tan/9  I  sinh  ^t\. 

That  is  to  say  \  i  sinh  t  tan  /3  +  (cosh  i  —  1 )  j  exceeds  botJt,  j  cosh  ^  —  1  j 
and  also  tan /3  |  sinh  ht\. 

In  order  to  simplify  the  subsequent  anafysis,  it  is  convenient 
to  place  a  restriction  on  /S.  We  shall  coiisequently  assume  in 
futui^e  that  O^/S^Itt,  so  that  tan^^^l.  This  restriction  is  not  of 
importance  so  far  as  the  final  result  is  concerned,  because  Debye's 
formula,  quoted  in  |  1  (ii),  is  effective  whenever  sec/3^1  +  S, 
where  8  is  any  positive  constant ;  and  so  it  is  certainly  effective 
when  sec  /3  ^  \/2.  The  importance  of  the  analysis  in  the  present 
investigation  is  due  to  the  fact  that  it  is  valid  for  small  values 

of  ;8. 

10.  Consider  what  happens  when  r  ^  ^,  whether  v,  V  are  both 
positive  or  both  negative. 

When  I  2'|  ^f ,  we  have  (on  the  T-contour) 

T  =  I iiTHanyS  +  irT' \  ^\T'\{^  +  ^\T\)<  h, 
and  if  I  i  I  <  I,  we  have  (on  the  ^-contour) 
T  =  I  [i  tan  /3  (cosh  ^  —  1)  +  (sinh  t  —  t)]\ 

00 

$  t  |«|'"//?i!^e*-l -1  =  2-12 -1-75  <^. 

Hence,  when  t  ^  |-,  we  must  have  both  |  ^j  ^  f  and  \t\  ^  f . 
But,  when  |  T  |  ^  | ,  we  have 

I  (dr/dT)  \  =  \T\.\itan/3+iT\^\T\.\^R(T)\^^\T\"-^^\. 
Also  (as  in  §  4)  when  |  ^  |  ^  f ,  we  have 
j  (dr/dt)  \  =  \i  sinh  t  tan  /3  4-  (cosh  t—  1)\ 

^  I  cosh  ^  —  1  I  =  cosh  u  —  cos  v^2  sin-  (fjr  ^2)  =  0-137, 

and  so  j  (dt/dr)  |  ^  7-3. 

From  these  results  we  see  that,  when  r^^, 

\  d  (t -T)/dT\<  15  <57r. 

We  shall  make  use  of  this  inequality  in  §  12. 

11.  Consider  next  what  happens  when  0  ^t  ^^,  whether  v,  V 
are  both  positive  or  both  negative. 

When  \T\^2,  we  have  (on  the  IT-contour) 

Also,  when  \t\  ^2  and  v  +  /3  ^  ^ir,  we  have  u  ^  ^tt \/3,  and 
then 

T  =  u  —  sec  /3  sinh  u  cos  (v  +  /3)^u^  1. 

H_-2 


108  Mr   Watson,  Bessel  functions  of  large  order 

Next,  when  |  ^  |  ^  2  and  ^  <^r  +  jS -^^Tr,  v/e  have 
cosh  w  =  (sin  0  +  v  cos  yS)  cosec  (v  +  0) 

^sm/3  +  (Itt  —  /3)cos/3  <  ^7r<  cosh  1*1, 

since  sin  yS  +  (^tt  —  ^)  cos  /3  is  a  decreasing  function  of  y8. 

This  gives  2  ^  |  i  |  <  \/{(-^7r)-  +  (1"1)^}  <  \/3'7,  which  is  impossible; 
so  that,  when  \t\^2,  we  cannot  have  /3  ^v  +  /3  -^  ^tt. 

Lastly,  when  \t\'^2  and  0  ^  v  ^  -  /3,  we  have  w  :$  0,  and  so 

-  w  ^  V(4  -  /S'O  ^  V{4  -  (iTT)}^  >  1-8, 
and 

T  =  sec  /3  sinh  (—  u)  cos  (?;  +  /3)  —  (—  it) 

^  sinh  (-  w)  -  (-  u)  >  ^  ( l-8)»  >  i . 

Therefore,  whenever  |  ^  |  ^  2,  we  have  r^^. 
Hence,  when  0  ^  t  ^  |- ,  we  must  have  both  \t\^2  and  |  T  |  ^  2. 
Next  we  shall  shew  that  R[^t+  T'^/{T  + 1)}  has  the  same  sign 
as  u  and  U. 

The  function  under  consideration  is  equal  to 

l^u  {{U  +  uf  +  (F+  vy\  +  (U"^-V')(U+u) 

+  2UV{V  +  v)]^[{U  +uy  +  (V  +  vf]. 

Taking  U,  V,  u,  v  positive  for  the  sake  of  definiteness,  we  see 
that  the  numerator  of  this  fraction  exceeds 

lu{U'  +V')  +  u{U-'- F^)  =  ^u(SU' -  V)  ^  0. 

Similarly  we  can  prove  that  the  numerator  is  negative  when 
U,  V,  u,  V  are  all  negative.     It  follows  from  this  result  that 

\R{t+Ty(T  +  t)}\^^\R{t)\^l\tl 

We  ewe  now  in  a  position  to  obtain  an  upper  bound  for  \1'  —  t\ 
when  1 1 1  and  \  T  \  are  both  less  than  2. 
First  suppose  that  |  ^  |  ^  | . 
Then,  from  the  formula  quoted  at  the  beginning  of  §  9, 

j  {T-  t)  I  .\{T+t)\.\  {laan  /3  +  i^  +  iTy{T+t)}  | 

=  I  i  tan  y8  (cosh  t-1-  lt~)  +  (sinh  t~t-lt^)\ 

00 

<   2    \t\'"'lm\^D\t\*lllQ. 

1)1  =  4: 

But  \T+t\^\t\and 
\{^its.n^  +  it  +  iTy{T+t)}\>i\R{t  +  ri(T+t)}\^^^^\t\. 

Hence,  when  |  i  |  ^  l,  we  have  |  (T  -  ^  |  ^  120  |  ^  |7119. 

Next,  keeping  |  ^  |  ^  | ,  we  take  the  formula 
i2iT-t)(T+t){iUnl3  +  i{T+t)} 

=  -^\(T-ty  +  i  tan  13  (cosh  ^  -  1  -  i  t~),+  (sinh  t-f-^f) 


3Ir   Watson,  Besael  functions  of  large  order  109 

and  observe  that 

and  also,  in  view  of  the  fact  that,  as  t  varies  through  positive 
vahies,  t  +  T  traces  out  in  the  Argand  diagram  a  curve,  through 
the  origin,  whose  slope  obviously  never  exceeds  V3,  the  distance 
of  all  points  of  this  curve  from  —  4^  tan  $  must  exceed  2  tan  /3. 
Hence  |  i  tan  /9  +  ^  (T  +  ^)  |  ^  ^  tan  /3. 

Using   these   two  inequalities,  combined  with   the   fact    that 
j(T— i)  j  ^  120|^|-/119,  and  the  obvious  inequalities 

\T+t\^\t\,     I  cosh  t-l-^t'\^\t  1 V23, 
|sinhi-^-i^-|^|^i-Yll9, 

we  deduce  from  the  last  equation  for  T— ^  that 

\T-t\^l\t\  {120  1 1  \ll\^Y  +  4 1  ^  IV23  +  16  U IV119  <  U  I'- 

Using  now  the  inequality  \T  —t\i^\t\^  in  place  of 

I  T- ^1^120  1^17119, 
we  get 

\T-t\^l\t\'  +  ^\t  IV23  +  16  I  ^  IV119 

<  (1/24  +  4/23  +  16/119)  \tY  ^  ^\t\K 

Using  now  the  inequality  \T  —t\^\\t\^,yNQ  get,  in  place  of  the 
last  result, 

\T-t\%  (1/192  +  4/23  +  16/119)  |  ^  1=^  ^  ^  |  ^  p'. 

From  this  result  it  follows  that,  when  l^j^^,  \T  —t\%  ^\t\, 
and  so  iri^l^l^l. 

Consequently,  from  the  formula  for  d{t  -  T)ldT  given  at  the 
beginning  of  §  9,  we  see  that,  when  |  ^  |  $  |^, 

\dt^_dJ 
dr      dr 


i\t\ 


{i|«|.|(cosh«-l)| 


+ 


i  (ii  I  ^  l)N  1*  ^i^h  ^  ^^^  /^  +  (cosh  t  —  1)| 
Now,  when  |  ^  |  <  2, 

1  po«?h  /_l|>i|^|2ri_     4     _     16     _  1>1|/|2 

and  \smh.t\^\t\[l-l;-^-  ...]>^\t\; 

and  so,  using  the  results  of  §  9,  we  get 

\d(t-  T)ldr \  ^  16/11  +  (576/121) [4 (1/6  +  1/23)  +  6/5] 
<12, 

when  1^1  ■S^.  . 


lio 


Mr   Watson,  Bessel  functions  of  large  order 


Lastly,  when  |:^|^|^2,  we  have  l^^j^  11/24,  and  so,  by  the 
method  of  §  10,  we  get 

\d{t-  T)ldr  I  <S  4  (24/11)^  +  \  cosec^  (i  ^2) 

<35-3<127r. 

12.     It  follows  from  the  results  of  §§  10, 11  that,  for  all  positive 
values  of  t, 

\d{t-T)ldT\<12Tr, 

and  consequently 


+ 


di 


<  24<7rjn, 


0  ;  [dr      dr]     '^ 

so  that 

ir,,«  (n  sec  y8)  =  A  e''^  (t=i»^-^)  g-"-  dT  +  24^6.,/ n, 

TJ"*  J-Qo-itan/3 

where  |  ^al  <  1- 

To  evaluate  this  integral,  where  —  r  =  ^T'^  i  tan  /8  +  ^T'^,  we 
take  the  contour  to  consist  of  the  two  rays  arg(T+ itan/8)  =  7r, 
^ir  ;  on  writing  T=  —  i  tan  /3  —  ^,  —i  tan  j3  +  ^e^"'  on  the  respective 
rays,  expanding  the  integrand  in  jDowers  of  ^  and  integrating  term 
by  term  we  find  that 

/■ooexp(47ri) 

e-''"dT 

J  -00  — ?!tan/3 

=  §771  tan  ^  exp  (— l^vn  tan^ /3) 

X  [e" *'''  J_  1  (i?^  tan^*  ,8)  +  e*"  /i  (iw  tan^ ^8)] 
=  3" -7ri  tan  /5  exp  (|■7^^  —  |-7w  tan^/3)  i/^.'^'  {^n  tan"  y8). 
Since  J„  {n  sec  ^)  =  R  [^w"'  ('^  sec  /3)], 

/_„  (w  sec  ^)  =  R  [e"'^^'  i^,,"'  ("  sec  ^8)] , 
it  follows  at  once  that,  when  0  ^  /3  ^  ^tt, 
Jn  {n  sec  /9)  =  3~^  tan  /3  cos  {m  (tan  /3  —  ^  tan^  /3  —  /3)} .  [JL  i  +  t/i] 

+  3" Han  /5  sin  {n  (tan  /3  -  i  tan^* /Q  -  /3)} .  [/_ .^  -  J^J  +  24(9/7^, 

J"_,i  (?i  sec  /3)  =  3~^  tan  /3cos  [ji  (tt  +  tan  /3  —  -^  tan'*  y8  —  /S)} .  [/_ x  +  Ji] 

+  3" ^  tan  /3  sin  {n  (tt  +  tan  /3  -  i  tan=*  ^  - /3)} .  [J_  j  -  ^^]  +  245'77i, 

where  the  arguments  of  the  Bessel  functions  J±x  on  the  right  are 
all  equal  to  ^ntan^/3,  and  |  ^  I,  \6'\  are  both  less  than  1.  It  is  easy 
to  see  that,  except  near  the  zeros  of  the  dominant  terms  on  the 
right,  the  ratios  of  the  error  terms  to  the  dominant  terms  are  of 
orders  Vl^^^^an^),  n~'^,  ?i~*,  according  as  7i  tan^ /3  is  large,  finite 
or  small. 


Mr  Todd,  A  particular  case  of  a  theorem  of  Dirichlet     111 


A  particular  case  of  a  theore^n  of  Dirichlet.  By  H.  Todd,  B.  A., 
Pembroke  College.  (Communicated,  with  a  prefatory  note,  by 
Mr  H.  T.  J.  Norton.) 

[Received  14  June  1917.] 

[The  following  note  is  an  extract  from  an  essay  submitted  to 
the  Smith's  Prize  Examiners. 

It  will,  perhaps,  be  convenient  if  I  preface  Mr  Todd's  argument 
by  explaining  its  relation  to  the  theory  of  algebraical  numbers. 
The  principal  theorem  is  a  famous  one  of  Dirichlet's  on  the  unities 
of  an  algebraic  corpus  or  order.  It  will  be  remembered  that  if  ^ 
is  a  root  of  an  irreducible  equation  of  the  nth  degree,  the  coefficients 
of  which  are  integers,  then,  if  the  coefficient  of  the  nth.  power  of 
the  unknown  is  1,  ^  is  an  algebraic  integer,  and  if  in  addition 
the  absolute  term  is  +  1,  ^  is  a  unity ;  and  further,  that  if  ^  is  an 
integer  of  the  ?ith  degree,  then  the  order  of  '^  is  the  aggregate  of 
numbers  w  of  the  form 

JUq  ~r~  ^1  ^j  *T   •  •  •  *^"ii 1  'J         ) 

where  x^... x^-i  are  rational  whole  numbers,  every  member  of  the 
order  of  ^  being  an  integer  of  the  wth  or  some  loAver  degree. 
Dirichlet's  theorem*,  as  modified  by  Dedekind  and  others,  asserts 
that  if  the  irreducible  equation  satisfied  by  ^  has  r  real  and  2s 
imaginary  roots,  then  the  order  of  ^  contains  r  +  s  —  1  fundamental 
unities,  e^,  ....  e,.+^.._i ,  which  are  such  that  every  unity  contained  in 
the  order  is  expressible  in  one  and  only  one  way  as  a  product 

'      ^  r+s-l' 

M'here  t;  is  a  root  of  unity  contained  in  the  order  and  m^, ... ,  m,.+.,_i 
are  rational  integers ;  and  that,  conversely,  every  such  product 
is  a  unity  and  a  member  of  the  order.  The  simplest  cases  of 
this  theorem  are  those  in  which  the  equation  satisfied  by  "^  is 
(i)  a  quadratic  with  two  imaginary  roots,  (ii)  a  quadratic  with  two 
real  roots,  (iii)  a  cubic  with  one  real  and  two  imaginary  roots  and 
(iv)  a  quartic  of  which  all  the  roots  are  imaginary.  In  the  first 
case,  and  in  this  alone,  there  are  only  a  finite  number  of  unities  in 
the  order,  and   they  are   all  roots   of  unity ;    in   the  other  cases 

*  The  theorem,  when  stated  completely,  has  a  wider  scope,  corresponding  to  a 
wider  definition  of  an  '  order '  than  is  given  above  :  what  is  there  defined  is  more 
properly  called  a  'regular  order'.  A  general  statement  and  proofs  are  given  in 
Bachmann,  Zahlentheorie,  vol.  v.,  eh.  8. 


112  Mr  Todd,  A  'particular  case  of 

mentioned  there  is  one  and  only  one  fundamental  nnit}'^  and  in 
cases  (ii)  and  (iii)  +  1  are  the  only  roots  of  unity  which  the  order 
contains.  In  case  (i)  the  theorem  is  easy  to  prove.  In  case  (ii), 
if  P  +  2bt  +  c  =  0  is  the  equation  satisfied  by  "^j  the  unities  of  the 
order  are  essentially  the  same  as  the  solutions  of  the  Pellian 
Equation 

x""  -  (b- -  c)  y^  =  ±  1, 

and  Dirichlet's  results  can  be  deduced  from  the  theory  of  this 
equation.  In  other  cases  the  proof  of  the  theorem  is  much  more 
difficult.  Mr  Todd  is  concerned  with  the  case  in  which  ^  is  the 
cube  root  of  an  integer — which  comes  under  the  heading  (iii) 
above.  If  ^  =  n,  the  general  theorem  ass( -rts  (a)  that  the  order 
of  ^  contains  an  infinity  of  unities,  (b)  that  they  are  all  expressible 
in  the  form 

where  7  is  a  particular  one  among  them  and  m  is  a  positive  or 
negative  whole  number,  and  (c)  that  every  number  of  this  form  is 
a  unity  of  the  order.  Mr  Todd's  essay  contained  an  elementary 
proof  of  (6)  and  (c) ;  the  proof  of  (c)  does  not  essentially  differ  from 
that  given  in  text-books,  though  this  was  not  known  to  him  at 
the  time,  but  the  proof  of  (6)  appears  to  be  new  and  forms  the 
subject  of  the  following  note. — H.  T.  J.  N.] 


If  ^^  — - n,  and  T  =x  +  y^  +  2'^^  is  a  member  of  the  order  of  ^, 
then 

r^  =  nz  +  x"^  +  2/^2^ 

SO  that  r  satisfies  the  cubic  equation 

\x  —  t,      y,         z 

\    nz,    x  —  t,      y     =0 ; 

I    ny,       nz,    x  —  t 


*4 


hence  it  follows  that  F  is  a  unity  of  the  order  if  and  only  if  x,  y,  z 
satisfy  the  Diophantine  equation 

=  af^-  ny'  +  n^z^  —  Snxyz  =  ±1 (i). 

It  will  be  the  object  of  this  short  note  to  give  a  simple  elemen- 
tary proof  of  the  fact  that,  if  the  existence  of  unities  is  assumed, 
then  every  unity  of  the  order  of  ^  can  be  expressed  in  the  form 


X, 

y, 

z 

nz, 

X, 

y 

ny> 

nz, 

X 

a  theorem  of  Dirichlet  113 

where  V  is  one  particular  unity  of  the  order,  and  m  is  a  positive 
or  negative  integer  or  zero. 

In  what  follows  we  shall  restrict  ourselves  to  the  positive  sign 
on  the  right-hand  side  of  equation  (i),  since  the  negative  sign 
merely  replaces  {x,  y,  z)  by  (—  x,  —  y,  —  z).  Also  when  x,  y,  z  are 
all  positive,  we  shall  refer  to  {x  +  2/'^  -f  s^-)  as  a  "  unity  of  positive 
integers  ". 

Suppose  that 

r  =  *•  +  2/^  +  z"^" 
is  any  unity  of  the  order  of  '^  :  we  shall  first  prove  the  following 
inequalities,  viz. : 

|a--2/^|,  JT/^-^^-j,  |^^-^-a;!^2/V(3r) (ii). 

For,  if  we  write  a-^-  x  —  y*^, 

^  =  y^-z^\ 

and  7  =  z^-  —  X, 

we  see  that  the  equation  satisfied  by  x,  y,  z  can  be  thrown  into 
the  form 

r(a^-h/3-^  +  7^)=2: 

so  that  we  have 

a-^+/3^  +  7'  =  2/ri 

and  a  +  /3  +  7  =  0     ]' 

From  these  two  equations,  assuming  F  to  be  constant,  we  find 
that  the  maxima  and  minima  for  each  of  a,  /8,  7  are 

±  2/V(3r)  ; 

from  which  the  truth  of  the  statement  (ii)  follows  immediately. 

Further,  we  have  the  fact  that  if  F  =  a;  +  2/^  +  z^"  is  any  unity 
of  the  order  and  F>  1,  then  x,  y,  z  will  be  positive. 

For,  since  F>  1,  we  have  the  inequalities 

\x-y'^\,\y'^-z'h^\,\z'^^-x\<  2/V3 <  US. 

But,  r  being  positive,  the  only  possibilities  of  negative  signs 
occurring  amongst  x,  y,  z  are  either  (a)  one  negative  and  two 
positive  or  (6)  two  negative  and  one  positive ;  and  in  each  case 
two  of  the  inequalities  given  would  take  the  form 

I  X,  +  yU<^  I  <  115, 

where  \  and  fi  are  positive  integers  and  <^  ^  \/2,  which  is  obviously 
impossible,  except  in  the  trivial  case  of  one  or  more  of  the  quantities 
x,  y,  z  vanishing  :  it  will  be  seen,  on  examining  the  inequalities, 
that  the  only  possibility  is  x  =  l,  y=0,  z  =  0,  which  gives  r  =  l 
and  so  is  excluded.     Hence  x,  y,  z  must  be  positive.     From  this 


114  Mr  Todd,  A  particular  case  of 

we  can  easily  shew  that  if  there  exists  an}^  unity  in  the  order 
other  than  +  1,  then  there  exists  a  unity  of  positive  integers  other 
than  +  1  of  which  any  other  unity  of  positive  integers  is-  a  positive 
integral  power.  For  suppose  that  T  is  any  unity  of  the  order 
other  than  +  1 :  then  by  definition  of  a  unity  it  follows  that  the 
three  numbers 

-r,    i/r,   -i/r, 

will  be  unities  of  the  order  also :  and  of  these  four  it  is  plain  that 
one  will  be  positive  and  greater  than  1,  i.e.  it  will  be  a  unity  of 
positive  integers. 

Now  take  any  number  k>  1;  then  there  will  be  only  a  finite 
number  of  F's  for  which  k  >T  >1,  since  for  any  such  F  we  must 
have  kXoO,  K>y>0,  k>  z>0.  Hence  there  must  be  a  unity 
of  positive  integers  which  is  greater  than  + 1  and  less  than  any 
other ;  let  this  one  be  7. 

Suppose  that  F  is  any  unity  of  positive  integers  which  is,  if 
possible,  not  a  positive  integral  power  of  7.  Then  we  shall  have 
F  >,7,  so  that  we  can  assume  that  F  is  intermediate  in  magnitude 
between  7^  and  7^+S  where  p  is  some  positive  integer.  But  by 
the  last  part  of  Dirichlet's  Theorem  we  know  that 

F/7^        - 

is  also  a  unity  of  the  order,  i.e.  we  have  found  a  unity  of  the  order 
which  is  less  than  7  and  greater  than  + 1,  which  contradicts  the 
assumption  that  7  was  the  least  unity  greater  than  +  1.  Hence 
F  must  be  a  positive  integral  power  of  7.  Finally  Ave  have  the 
result  that,  if  F  is  any  unity  of  the  order,  it  can  be  expressed  in 
the  form 

where  7  has  its  previous  significance  and  jo  is  any  positive  or 
negative  integer  or  zero.  For  if  F  is  any  unity  of  the  order,  other 
than  +  1,  the  numbers 

-F,     1/F,     -1/F 

also  will  be  unities,  and  one  of  these  will  be  positive  and  greater 
than  1,  and  so  will  be  expressible  in  the  form 


where  g-  is  a  positive  integer.     Hence  F  can  be  expressed  in  the 
form 

where  p  is  some  positive  or  negative  integer  or  zero. 

The  result  obtained  can  be  put  into  an  interesting  geometrical 
form  as  we  shall  proceed  to  shew. 


a  theorem  of  Dirichlet  115 

It  is  evident  that  any  rational  point  (*•,  y,  z)  in  space  of  three 
dimensions  can  be  regarded  as  being  determined  by  its  affix 
V^x-]ry^  +  z^'\  where  ^  is  the  real  root  of  the  equation  ^^  =  n  : 
also  the  affix  of  any  point  determines  a  plane  through  that  point 
and  parallel  to  the  asymptotic  plane  of  the  surface  whose  equation 
is  i\=(jc?  -\-  ny"^  +  n"z^  —  Snxyz  =  1  ;  such  a  plane  we  shall  call  a 
"  r-plane  ". 

We  shall  now  prove  the  following  proposition : 

The  V-planes  of  any  two  consecutive  integj^al  points  on  the 
surface  A  =  1,  together  with  the  surface  itself,  enclose  a  space  of 
constant  volume. 

The  equation  A  =  1  can  be  written  in  the  form 

[x  +  y^  +  2^-|  [{x  -  y'^y  +  (2/^  -  z^^-'f  +  {z"^^  -  xy^  =  2  ; 

so  that  the  section  by  the  F-plane  of  the  point  (^,  rj,  f)  will  be 
given  by  the  equations 

x"  +  2/-^2  ^  ,^^^.^2  _  ,^^y^  _  ^2^^.  _  ^^.y  ^  ijY (i) 

and  a;  +  2/^  +  z"^'-  =  T. 

Evidently  the  quadric  (i)  and  the  surftxce  A  =  1  are  cut  in  a 
common  section  by  the  F-plane  of  the  point  (^,  77,  ^).  It  is  this 
quadric  that  we  shall  now  examine. 

If  by  any  rotation  of  axes  it  becomes  ax-  +  hy^  +  6'^^=  1,  we 
shall  have  (from  the  usual  properties  of  invariants) 

a4-6  +  c=r(l+  7i^  +  ^-),       \ 

ah  +  bc  +  ca  =  ^  P^-  (1  +  m^  +  ^-),  I 

abc  =  0  ;  j 

so  that  the  quadric  is  evidently  a  cylinder,  and  the  direction  of  its 
axis  is  the  line  x  =  y^  =  z^". 

Suppose  that  c  =  0 ;  then  the  area  of  a  right  section  of  the 
cylinder  will  be 

-rr/^/iah)  =  |^/V3  (1  +  w^  +  ^^). 

But  the  angle  between  the  normals  to  the  right  section  and  the 
F-plane  is  the  same  as  the  angle  between  the  two  lines 

and  x  =  y/'^  =  zl'^^; 

i.e.,  is  cos-i  {3^7(1  +  n^  +  ^-)} : 


116     il/r  Todd,  A  partictdar  case  of  a  theorem  of  Dirichlet 
hence  the  area  of  the  section  made  by  the  F-plane  will  be 
27r  V(l  +  «^  +  ^-)/3nV3  F. 

Now  the  perpendicular  distance  between  two  near  F-planes, 
r  and  r  +  ST,  is  8r/V(l  +  >i^  +  ^-),  and  so  the  element  of  volume 
enclosed  by  these  two  planes  and  the  surface  A  =  1  will  be,  to  the 
first  order, 

27r      ST 
3nV3'^' 

Integrating  this  between  the  limits  F  =  7^+^  and  F  =  7^'  (i.e.  the 
F-planes  of  any  two  consecutive  integral  points),  we  find  that  the 
volume  of  the  space  enclosed  is  27r  log  7/3/1^3 ;  and  since  this  is 
independent  of  the  integer  p,  our  proposition  is  proved. 


a 


Mr  Mordell,  On  Mr  Ramanujans  Empirical  Expansions,  etc.     117 


On  Mr  Ravianujan's  Empirical  Expansions  of  Modular 
Functions.  By  L.  J.  Mordell,  Birkbeck  College,  London.  (Com- 
municated by  Mr  G.  H.  Hardy.) 

[Received  14  June  1917.] 

In  his  paper*  "On  Certain  Arithmetical  Functions"  Mr 
Ramanujan  has  found  empirically  some  very  interesting  results 
as  to  the  expansions  of  functions  which  are  practically  modular 
functions.     Thus  putting 

(^X^{<o„  CO,)  =  r  [(1  -  r) (1  - rO (1  - r^)  . ..?  =  S  T {n)  r-, 

he  finds  that 

T{mn)  =  T(m)T{n)  (1) 

if  m  and  n  are  prime  to  each  other ;  and  also  that 

2  ^^  =  Ul/(l-T(p)p-^+p^^-) (2), 

n=l      "' 

where  the  product  refers  to  the  primes  2,  3,  5,  7 He  also  gives 

many  other  results  similar  to  (2). 

My  attention  was  directed  to  these  results  by  Mr  Hardy,  and 
I  have  found  that  results  of  this  kind  are  a  simple  consequence 
of  the  properties  of  modular  functions.     In  the  case  above 

A  (&)i,  Wa)      (r  —  e'"'^"",   (o  =  coi/coj) 

is  the  well-known  modular  invariant  (^f  dimensions  —  12  in  co^,  0)2, 
which  is  unaltered  by  the  substitutions  of  the  homogeneous 
modular  group  defined  by 

ft)/  =  aoii  +  bco.2,  (o./  =  Cftji  +  dw2, 

where  a,  b,  c,  d  are  integers  satisfying  the  condition  ad  —bc  =  l. 

Theorems  such  as  T  {mn)  =  T  {m)  T  {n)  had  already  been 
investigated  by  Dr  Glaisher  f  for  other  functions ;  but  the 
theorems  typified  by  equation  (2)  seem  to  be  of  a  new  type,  and 
it  is  very  remarkable  that  they  should  have  been  discovered 
empirically.     The  proof  of  Mr  Ramanujan's  formulae  is  as  follows. 

Let  f{(Oi,  6)2)  be  a  modular^  form  of  dimensions  ~  k  in  coj,  co,, 
which  is  a  relative  invariant  of  the  homogeneous  modular  gi'oup, 
so  that  /(fw/,  w.^)l f{(o^,  0)2)  is  a  constant  independent  of  «i,  (Oo,. 

*  Transactions  of  the  Cambridge  Philosophical  Society,  vol.  xxii. ,  no.  ix. ,  1916. 

t  See,  for  example,  his  paper  "  The  Arithmetical  Functions  P  (m),  Q  (in),  fi  (;n)  ", 
Quarterlij  Journal  of  Mathematics,  vol.  xxxvii.,  p.  36. 

+  For  an  elementary  introduction  to  the  modular  functions,  see  Hurwitz, 
Mathematische  Annalen,  vol.  18,  p.  520, 


118  Mr  Mordell,  On  M^-  Ramanujan's 

Let  also  p  be  any  prime  number;  then  we  may  take 

(&)i,  pa>^,  («!  +  CDa,  p(o<^  . . .  (ft)i  +  ( jO  —  1)  fUo,  J9&)y),   (  ptWi ,  &),.) 

as  the  reduced  substitutions  of  order  j)-  Then  for  many  modular 
forms*  it  is  well  known  that  unities  ^,  fu,  ^i,  ••■,  |>-i  can  be 
found  so  that 

is  also  a  relative  invariant  of  the  modular  group. 

This  is  also  true  of  the  quotient  Q  =  (fi/fioy^Jwo),  which  is  a 
modular  function  of  co.  Q  is  really  an  automorphic  function  whose 
fundamental  polygon  (putting  (o  =  x  +  Ly)is  that  part  of  the  upper 
CO  plane  bounded  by  the  lines  x  =  ±^  and  external  to  the  circle 
a;2  +  2/2=  1,  but  we  reckon  only  half  the  boundary  as  belonging  to 
the  fundamental  polygon.  The  only  infinities  of  Q  are  given  by 
the  zeros  of /(&)i,  0^2)  =  0,  and  if  these  zeros  are  also  zeros  of  the 
numerator  of  at  least  the  same  order  as  of  the  denominator, 
it  follows  that  Q  has  no  infinities  in  the  fundamental  polygon. 
Hence  Q  is  a  constant,  so  that  (f)~Qf((Oi,  &>.,). 

Suppose  now  that 

where  ^1  =  1.     Then 

becomes  (^)"  S's' |.^,r^/^e^-'VP 

and  in  the  examples  with  which  we  are  concerned  all  the  terms 
will  vanish,  because  of  the  summation  in  X,  except  those  for  which 
s  =  0  (mod  p),  and  then  the  sum  will  become 


^  pA, 


Hence  we  have 


Equating  coefficients,  we  find,  if  s  is  prime  to  p, 
pAsp  =  Qp^Ag. 

*  This  fact  is  intimately  connected  with  the  transformation  equations  in  the 
theory  of  the  modular  functions.  We  may  note  that  it  is  often  more  convenient  to 
select  the  reduced  substitutions  in  different  ways. 


Empirical  Expansions  of  Modular  Functions  119 

Taking  .9=1,  pAp  =  Qp", 

so  that  -Agp^  AsAp... (3). 

If  no  restrictions  are  placed  on  s  we  find,  by  equating  coefficients 
of  rP', 

^^s+^,Asp.=  QAsp. 

From  this 

Asp.-ApA,p  +  ^p''-'A.  =  0 (4). 

From  equations  (3)  and  (4),  we  can  prove  that  A^n  =  A^An 
if  m  and  n  are  prime  to  each  other.  For  all  we  really  have  to  shew 
is  that,  if  2?  is  a  prime  and  s  is  prime  to  j),  then  ^,,^a  =  AifAp\. 
But  from  equation  (4),  we  have 

Asp\+2  -  ApAspK-hi  +  ^p"-^  Agp\  =  0, 

and  Ap\+2  —  ApAp\+i  +  ^j)''-^  A^k  =  0 (4a). 

Hence  the  theorem  follows  by  induction,  for  if  it  is  true  for  \ 
and  X  + 1  it  is  true  for  X,  +  2.  But  it  is  true  for  X  =  0  and  for 
X=  1  (equation  3):  hence  it  holds  universally. 

We  notice  also  that  equation  (4a)  is  a  linear  difference  equation 
of  the  second  order  with  constant  coefficients*.  Hence,  since 
A,=  l, 

1  +  ApX  +  Ap-iX"  4  Ap^x^  +  . . .  =  1/(1  -  ApX  +  ^p^-Kx-), 
from  which,  by  putting  x  =  l/jf, 

pS  p2S  p3S    '^   ••■  /    ^  p'  p^^    J' 

Putting  for  p  in  succession  the  primes  2,  3,  5  ...,  multiplying 
together  the  corresponding  equations,  and  remembering  that 
A,nn  =  A^nAn  if  vi  and  /;  are  prime  to  each  other,  we  have 

« 

where  the  product  refers  to  the  primes  2,  3,  5  — 

The  simplest  application  of  these  results  is  given  by  the 
function 


fa  {co„  &).3)  =      A  ^—  ft),,  (Ooj 


This  is  obvious  if  we  put  fx^=  ApK. 


120  My^  Mordell,  On  Mr  Ramanujan's 

where  a  is  a  divisor  of  12.     Its  expansion  in  powers  of  r  involves 

only   positive   integral    powers    of  r    and    starts   with    I  —  j  r. 

/a(&)i,  6)2)  is  not  however  an  invariant  of  the  modular  group. 
We  can  avoid  this  difficulty  by  taking /(co,,  eo,)  =  [A(ft)i,  w^)]"'^'^. 
In  this  case* 

Kp-ni 

^f(pco^,a),)  =L(-1)   ^    iyA(p&)i,  &)2)J  , 

provided  we  exclude  p  =  2  and  p  =  S.     Putting  for  the  moment 


/m    \"       / "=  —4-'! 

we  find 


(—:)   [iyA(a,„a,,)]-  =  S^^,ri-^^^ 


CO     p-i         OKpni     (  a       \  2K7ri  /■  «■   ,  „N  1 


2 

K  =  0 

=  22  e"~6 

S  =  0/<:  =  0 

But  since  ^  4"  ^  or  3,  p'  —1=0  (mod  12).     Hence 
2  eP  i    12   +*;=o, 

K  =  0 

unless  a{\  —p-)/l2  +  s  =  O(mod_p),  that  is  a 4-  12s  =  0  (mod  jd),  and 
is  then  equal  to  p.  Hence  ^  is  a  power  series  in  r^^^^  (really  of  the 
form  ?-'^/i2(^  +  Br+  Gr"  ...)),  starting  with  r'<"'+'^^s)i\ip^  where  s  is 
the  smallest  positive  integer  for  which  a  +  12s  =  0  (mod  p).  Now 
the  only  zeros  of /(wi,  oa^)  =  0  in  the  fundamental  polygon  are  at 
ft)  =  iX)  or  r  =  0,  and 

/(o)i,  «2) = (^y '■'"'  (1 + ^^' +^^ ...). 

But  putting  a  =  1 2/6,  so  that  h  is  an  integer, 


a  +  ]  2s     1  +  6s     1      a 


12p  bp     "6^12' 

since  1  +  6s  =  0  (mod  p). 

Hence   <^lf{ui^,  Wa)  is  a  constant,  and  equations  (8),  (4),  (5) 
apply  to  the  function 

V  a 
We  note  also  that  |=  (-  l)«(^-i)/2. 

*  Hurwitz,  I.e.,  p.  572,  or  Weber,  Lelirhuch  der  Algebra,  vol.  3,  p.  252 


i 


«/12 


Empirical  Expansions  of  Modular  Functions  121 

When  p  =  2,  these  theorems  hold  if  a  =  4  or  12.  For  the 
functions  ^\f\*  are  selected  as  before,  and  it  is  clear  that  the 
argument  above  applies,  as  « (1  —p^)l\2  is  an  integer. 

Lastly,  when  p  =  S,  these  theorems  hold  if  a  =  3,  6,  12,  and  the 
functions  ^k/k*  are  selected  as  before. 

Hence,  altering  our  notation,  we  have  the  following  theorems. 
If  a  is  a  divisor  of  12  and 

p  12  24  36  -j,,^         00 

r  [(l  -r")  (l-r«)  (l  -r")  .-..J^"  =  S  fa(n)r'\ 

n  =  \ 

then  fa  {m)fa  (n)  =fa  (mn) (6), 

if  m  and  n  are  prime  to  each  other ;  and 

^  />- (^) ^ n  1  /(i  -^^ ^^^  I  ^~ ^K^^''~'] (7). 


n=i    n'  /  V  p'  P~ 

The  product  refers  to  the  primes  2,  3,  5,  etc.,  except  that 

p  =  2  is  excluded  except  when  «  =  4,  12, 
and  jt)  =  3  is  excluded  except  when  a  =  .3,  6,  12. 

We  notice  that  when  a  —  1,  2,  3,  or  6,  jt?  =  2  is  not  excluded 
as  a  factor  of  say  m  in  (6),  as  in  this  case  fai'ni)  and  fa  (mn)  are 
both  zero.     Similarly  for  jo  =  3  when  a  =  1,  2,  4. 

The  result  (6)  is  given  by  Mr  Ramanujan  wh6n  a  =  12,  as  are 
most  of  the  cases  of  (7).  We  shall  now  shew  how  in  many  cases 
we  can  find  simple  expressions  for  fa{p). 

If  a  =  1,  it  is  known  that,  by  a  result  due  to  Kulerf, 

r[(l-?-i2)(l-r'^^)...]2  =  [  2  (-If  ?•      2     J 

—  00 

(6ot  +  1)2+(6w  +  1)2 

=  SS(-l)'"+"r  2 

=  tt(-l)^r^'^^^\ 

where  ^  =  3  (m  +  ?i)  +  1,  r)  =  n  —  m,  so  that  ^,  ij  take  all  integer 
values  satisfying  ^  =  1  (mod  3),  ^  +  77  =  1  (mod  2). 

Hence  f(p)  —  2(—  1)''  if  p  =  ^^  +  9r}^  and  we  take  both  ^  and  77 
to  be  positive.  If  ^  =  —  1  or  ±5  (mod  12),  f{p)  is  obviously  zero. 
This  is  Mr  Ramanujan's  result  (118). 

If  a  =  2,  it  is  known  (Klein-Fricke,  vol.  2,  page  374)  that 

r  [(1  -  r«)  (1  -  r'^)  ...]*  =  ^2  (-  1)^  ^r^'+^^^+^r,^, 
where  f ,  r)  take  all  integer  values  satisfying 

1=2  (mod  3),   7;  =  1  (mod  2). 

*  Hurwitz,  I.e.,  vol.  18. 

t  See  also  Klein-Fricke,  Modulfunktionen,  vol.  2,  p.  374. 

VOL.  XIX.  PARTS  II.,  III.  9 


122  Mr  Mordell,  On  Mr  Ramanujans 

Hence  /.(jt>)  =  2  i(- 1)^| 

extended  to  the  solutions  of  p=  ^  +  S^rj  +  Srj-  for  which 

^  =  2  (mod  3),   17  =  1  (mod  2). 

This  *  can  be  written  as  /2  (  p)  =  ^v,  where  p  =  Su^  +  v'-,  u  is  positive 
and  V  =  1  (mod  3).  Also  f\  {p)=0  if  p  =  -  I  (mod  3).  This  is 
Mr  Ramanujans  result  (127). 

If  a  =  3  we  have,  from  Klein-Fricke,  vol.  2,  page  377, 

r  [(1  -  r^)  (1  -  r«)  ...]«  =  -  12  (p  -  7)')  rf=+''^ 
where  ^  takes  all  even  values  and  rj  all  odd  values.     Hence 

if  p  =  ^^  +  7)"^,  I  is  even,  97  is  odd,  and  both  f  and  ■?;  are  positive. 
Also  /3( j9)=  0  if  ^  =  3  (mod  4).  This  is  Mr  Ramanujans  result 
(123). 

If  a  =  4,  then  by  Klein-Fricke,  vol.  2,  page  373, 

r  \(l  —  r^)  (1  —  r^) . . .?  =  +2p  r^""*"^^*''+^''', 

where  ^,  rj  take  all  values  for  which  |^  =  2  (mod  3). 
Hence /4  (^)  =  ^Sp  extended  to  all  the  solutions  of 

^  =  p  +  3^77  +  '67j\ 

where  |  =  2  (mod  3).  Thisf  can  be  written  as  /4  (  jd)  =  2  (v"  —  9vu-), 
where  p  =  3w'^  +  ^^  tt  is  positive,  and  y  =  1  (mod  3).  This  is 
Mr  Ramanujans  result  (128). 

When  a  =  6,/6  {n)  is  known  by  means  of  the  representations  of 
n  as  a  sum  of  four  squares.  Mr  Ramanujan  has  overlooked  the 
fact  that  in  his  result  (159)  2c^  is  —J\{p)'     The  theorem 

/b  (wO/e  in)  =/6  (mn), 

is  due  to  Dr  Glaisher. 

When  a  =  12,  we  have  Mr  Ramanujan's  results  given  as 
equations  (1)  and  (2)  in  this  paper. 

He  also  gives  results  when  a  =  i  ,  |. 

*  When  ^  is  even  put  ^  =  2v,  7]  =  u-v,  and  when  |  is  odd  put  ^  =  du-v,'r)  =  v-7i. 
Both  these  cases  are  admissible,  and  we  find  that  p=v^  +  Bu"  and  v  =  l  (mod  3). 
Also  S  {-1)^ ^=2v  +  2v  -  {Su  -v)  -  (  -3u-v)  =  Gv,  where  now  w  is  taken  as  positive. 

t  See  the  last  footnote.  In  addition  to  the  two  cases  there  considered,  7/  even 
is  admissible.  Put  then  7]  =  2u,  ^—  -v  -3ii,  from  which  p  =  v^  +  3ii'^  and  v  =  l 
(mod  3). 


Empirical  Expansions  of  Modular  Functions  123 

where   [- j  and   ( ]  are   symbols  of  quadratic   reciprocity,    so 

that   (:-^)  =(-l)V  ,  (I)  =  1  if  i,  EE  ±  1  (mod  12),  and  (|)  =  -  1 


if  jj  =  ±  5  (mod  12).    If  /;  =  -S,  ('^\  =  0. 

These  are  particular  cases  of  Euler's  theorem  that 

s-^^^^=ni/(i--^^^) 

if  the  function  y"  satisfies  the  condition 

f{m7i)=f(m)f(n), 

the  product  refers  to  any  group  of  primes,  and  the  summation  to 
all  numbers  whose  prime  factors  are  included  in  the  group.    Thus 

r  (1  _  r2J)(l  _  r'')  ...  =  i  (_i)»y.(««+i)^=    v      f'}\  ,,.«^       . 

-c»'  1,3,  5...  V''^/ 

and     r  [(1  - ?•») (1  -  r^^)  ...]'=    t    (-  1) ^  wr^'  =    5      f ^  nr"\ 

1,3,5...  1,3,5...  \   n   J 

Finally,  Mr  Raman ujan  gives  two  results,  equations  (155)  and 
(162),  of  which  the  first  is 

5  -^  =  lT9i=^  n  1/(1  -  2c,  p-'  +  (-  IV   P'-n 

where  Cp  =  u^  -  (4w)-  and  u  and  y  are  the  positive  integers  satis- 
fying u^  H-  {^v)'=}f:  But  if  JO  =  3  (mod  4),  Cp  is  taken  to  be  zero, 
/lo  (w)  is  defined  *  by 

=^r[(l-r^)(l-r^)(l-r«)...]"/[(l+r)(l-r-^)(l-|-r^)(i-r^)...h 
and  this  is  equal  tof 

ii  2(a,-  +  i2/)^r^'+^'. 

—  00  —00 

The  second  result  is 

I'^^^lT^^W-Sc^p-^+i'^-'X     (1^  =  3,5...), 

"  The  functions /io(?i),/i6  (n)  arise  in  iinding  the  number  of  representations  of 
n  as  a  sum  of  10  and  16  squares  respectively  and  the  series  2  S  (a;  +  i?/)-*r^""'"*'  is 
well  known  in  this  connection. 

t  From  this,  it  follows  that  the  result  can  be  also  proved  as  a  particular  case 
of  Euler's  product. 


124     Mr  Mordell,  On  Mr  Ramanujan's  Empirical  Expansions,  etc. 
where /le  (?i)  is  defined*  by 

1 

=  r  [(1  +  r)(l-r2)(l  +rO(l  -r^) . ..]'/[(!  -r"){l-7^){\-r') ...]«. 

Mr  Ramanujan  overlooks  the  fact  that  Cy  =  ^fu{p)- 
These  results  can  be  proved  by  aid  of  the  principles  used  in 
finding  equations  (3)  and  (4).    We  should  however  have  to  consider 
now  invariants  of  a  sub-group  of  the  modular  group,  and  it  seems 
hardly  worth  while  to  go  into  details. 

*  The  functions /lo  ('O'/ieC**)  arise  in  finding  the  number  of  representations  of 
n  as  a  sum  of  10  and  16  squares  respectively  and  the  series  2  S  [x  +  njY  r*  "^^^  is 
well  known  in  this  connection. 


PROCEEDINGS   AT   THE   MEETINGS    HELD   DURING 

THE   SESSION    1916—1917. 

ANNUAL   GENERAL   MEETING. 

October  30,   1916. 

In  the  Comparative  Anatomy  Lecture  Room. 

Professor  Newall,  President,  in  the  Chair. 

The  following  were  elected  Officers  for  the  ensuing  year : 

President: 
Dr  Marr. 

Vice-Presidents : 

Dr  Fenton. 
Prof.   Eddington. 
Prof.   Newall. 

Treasurer  : 
Prof.  Hobson. 

tSecretanes  : 

Mr  A.   Wood. 
Mr  G.   H.  Hardy. 
Mr  H.  H.  Brindley. 

Other  Members  oj  the  Council  : 

Dr  Duckworth. 

Dr  Crowther. 

Dr  Bromwich. 

Dr  Doncaster. 

Mr  C.  G.   Lamb. 

Mr  J.   E.   Purvis. 

Dr  Shipley. 

Dr  Arber. 

Prof.  Biffen. 

Mr  L.   A.  Borradaile. 

Mr  W.   H.  Mills. 

Mr  F.  F.  Blackman. 


126  Proceedings  at  the  Meetings. 

The  following  was  elected  an  Associate  of  the  Society  : 
W.   Morris  Jones,   Emmanuel  College, 

The  following  Communications  were  made  : 

1.  Methods  of  investigation  in  atmospheric  electricity.  By 
C.  T.  R.  Wilson,  M.A.,  Sidney  Sussex  College. 

2.  On  the  functions  of  the  mouth  parts  of  the  Common  Prawn, 
By  L.  A.  BoRRADAiLE,  M.A.,  Selwyn  College. 

3.  On  the  growth  of  Daphne.  By  J.  T.  Saunders,  M.A.,  Christ's 
College. 

4.  A  self-recording  electrometer  for  Atmospheric  Electricity.  By 
W.  A.  D.  Budge,  M.A.,  St  John's  College, 

5.  An  axiom  in  Symbolic  Logic.  By  C.  E.  Van  Horn.  (Com- 
municated by  Mr  G.  H.  Hardy.) 

6.  On  the  expression  of  a  number  in  the  form  aar  -i-  hy-  +  cz^  4-  du". 
By  S.  Ramanujan,  Trinity  College.  (Communicated  by  Mr  G.  H. 
Hardy.) 

7.  A  reduction  in  the  number  of  primitive  propositions  of  Logic. 
By  J.  G.  P.  NicoD,  Trinity  College.  (Communicated  by  Mr  G.  H. 
Hardy.) 


November  13,   1916. 

In  the  School  of  Agriculture.  ^ 

Dr  Marr,  President,  in  the  Chair. 

The  following  were  elected  Fellows  of  the  Society  : 

F.  W,  Green,  M,A,,  Jesus  College, 
R,  I,  Lynch,  M.A. 

The  following  was  elected  an  Associate  of  the  Society  : 
N.  Yamaga,  Fitzwilliam  Hall, 

The  following  Communications  were  made  : 

1.  The  surface  law  of  heat  loss  in  animals.     By  Professor  Wood. 

2.  Inheritance  of  henny  plumage  in  cocks.     By  Professor  Punnett 
and  Capt.  P.  G.  Bailey. 


Proceedings  at  the  Meetings.  127 

3.  On  extra  mammary  glands  and  the  reabsorption  of  milk  sugar. 
By  Dr  Marshall  and  K.  J.  J.  Mackenzie,  M.A.,  Christ's  College. 

4.  Experimental  work  on  clover  sickness.     By  A.  Amos,  M.A., 
Downing  College.     (Communicated  by  Professor  BifFen.) 

5.  Bessel's  functions  of   equal  order  and  argument.     By   G.   N. 
Watson,  M.A.,  Trinity  College. 


February  5,   1917. 

In  the  Sedgwick  Museum. 

Dr  Marr,  President,  in  the  Chair. 

The  following  was  elected  a  Fellow  of  the  Society : 

F.  W.  H.  Oldham,  B.A.,  Trinity  College. 

The  following  Communications  were  made : 

1.  Submergence  and  glacial  climates  during  the  accumulation  of 
the  Cambridgeshire  Pleistocene  Deposits.     By  Dr  Marr. 

2.  Glacial  Phenomena  near  Bangor,  North  Wales.  By  P.  Lake, 
M.A.,  St  John's  College. 

3.  The  Cretaceous  Faunas  of  New  Zealand.  By  H.  Woods,  M.A., 
St  John's  College. 

4.  Exhibition  of  the  Fruit  of  Chocho  Sechium  edule :  remarkable 
in  the  Nat.  Order  Cucurbitaceae,  native  of  the  West  Indies  and  culti- 
vated also  in  Madeira  as  a  vegetable.     By  R.  I.  Lynch,  M.A. 

5.  The  limits  of  applicability  of  the  Principle  of  Stationary  Phase. 
By  G.  N.  Watson,  M.A.,  Trinity  College. 

6.  The  Direct  Solution  of  the  Quadratic  and  Cubic  Binomial 
Congruences  with  Piime  Moduli.  By  H.  C.  Pocklington,  M.A., 
St  John's  College. 

7.  On  the  Hydrodynamics  of  Relativity.  By  C.  E.  Weather- 
burn,  M.A.,  Trinity  j(^ollege. 

8.  The  Character  of  the  Kinetic  Potential  in  Electromagnetics. 
By  R.  Hargreaves,  M.A.,  St  John's,  College. 

9.  On  the  Fifth  Book  of  Euclid's  Elements.  (Fourth  Paper.)  By 
Dr  M.  J.  M.  Hill. 

10.  On  a  theorem  of  Mr  G.  Polya.  By  G.  H.  Hardy,  M.A. 
Trinity  College. 


128  Proceedings  at  the  Meetings. 

February  19,   1917. 
In  the  Botany  School. 

Dr  Marr,  President,  in  the  Chair. 

The  followiug  Communications  were  made  : 

1.  (1)     On  an  Australian  specimen  of  Clepsydropsis. 

(2)  Observations  on  the  Evolution  of  Branching  in  the 
Ferns.  By  B.  Sahni,  B.A.,  Emmanuel  College.  (Communicated  by 
Professor  Seward.) 

2.  On  some  anatomical  characters  of  coniferous  wood  and  their 
value  in  classification.  By  C.  P.  Dutt,  B.A.,  Queens'  College.  (Com- 
municated by  Professor  Seward.) 


CONTENTS. 

PAGE 

The  Direct  Solution  of  the  Quadratic  and  Cubic  Binomial  Congruences 
tuith  Prime  Moduli.  By  H.  C.  Pocklington,  M.A.,  St  John's 
College .57 

On   a  theorem   of  Mr   G.   Polya.     By   G.    H.    Hardy,    M.A.,    Trinity 

College      . 60 

Submergence  and  glacial  climates  during  the  accumulation  of  the  Cam- 
bridgeshire  Pleistocene  Deposits.  By  J.  E.  Maer,  Sc.D.,  F.R.S., 
St  John's  College       .         .         .         .         .         .         .         .         .         .64 

On  the  Hydrodynamics  of  Relativity.     By  G.  E.  Weatherburn,  M.A. 

(Camb.),  D.Sc.  (Sydney),  Ormond  College,  Parkville,  Melbom'ne      .       72 

On  the  convergence  of  certain  multiple  series.     By  G.  H.  Hardy,  M.A., 

Trinity  College .         .         .86 

Bessel  functions   of  large   order.     By   G.   N.  Watson,   M.A.,  Trinity 

College 96 

A  particular  case  of  a  theorem  of  Dirichlet.  By  H.  Todd,  B.A., 
Pembroke  College.  (Communicated,  with  a  prefatory  note,  by 
Mr  H.  T.  J.  Norton) Ill 

Oil  Mr  Ramanujan's  Empirical  Expansions  of  Modular  Functions.  Bj^ 
L.  J.  MoRDELL,  Birkbeck  College,  London.  (Communicated  by 
Mr  G.  H.  Hardy)  .         . .117 

Proceedings  at  the  Meetings  held  during  the  Session  1916 — 1917  .         ,     125 


PKOCEEDINGS 


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PROCEEDINGS 

OF   THE 


Extensioits  of  Abel's  'Theorem  and  its  converses.  By  Dr  A. 
KiENAST,  Kiisnacht,  Zurich,  Switzerland.  (Communicated  by 
Mr  G.  H.  Hardy.) 

[Received  26  September  1917.] 
Introduction. 
Abel  pruved  in  1826  the  theorem  : 

n 

''If  liui  !£  (^  ejists  and  is  finite,  then 


lim  %  a^x"  =  lim  ^  a„." 


Let  us  write 


Sn 

1 

(Ik 

t""' 

_1 

S6V 

n 

n 

'      ! 

^(A  +  1) 

_1 

i  (« 

n 

1                     / 

Then  H(3lder*  proved  in  1882 

Theorem  1.     If  lim  t^^    exists  and  is  finite,  then 


.(1). 


lim  S  ai^x"  =  lim  t]^  . 

*  Bromwifh,  Iiijinite  scries,  p.  313. 
VOL.  XIX.   PART  IV. 


10 


130 


Dr  Kienast,  Extensions  of 


In  1897  Mr  Tauber,  and  in  1900  Mr  Pringshciui,  published  the 
following  converse  of  Abel's  theorem  : 

Theorem  2.     The  two  conditions 

lim  2  ««.'■"  =  I     (  finite), 

1  " 

hm  -  2  KCi^  =  0 

n-*'Oo  ^^  1 

are  each  necessary  for  the  convergence  ofX  a^,  i.e.  for  the  existence  of 

n 

lim  ^a^  —  l; 

n-*-ao  1 

and,  taken  together,  they  are  sufficient'^. 

In  the  present  paper  I  replace  the  means  (1)  by 


^ (2.) 


1 

«          ?l     1 

{n  =  2,  3, . 

(«.  =  X  +  2, 

0^  =  1,2,. 
{n  =  2,  3, . 

•■) 

Defining  r\'^  by 

■) 
•) 

r;'+^^  =  V  -  7-(^)      Ol  =  X  + 1,  X  +  2, . . . ) 


I  prove,  in  Part  I, 

Theorem  3.     Tlie  tivo  conditions 


•(3), 


lim  S  a^x"  =  I     (finite), 

x-*l   1 
Broniwjch,  IntinlU  series,  p.  251. 


Abel's  TJieurcDi  and  its  converses  131 


are  each  necessary  for  the  existence  of 


lim  s^^^  =  / ; 


(tnd,  taken  together,  they  are  suMcient. 

This  theorem  includes  the  analogue  of  1  : 

Theorem  4.     If  lim  s\^  =1  exists  and  is  finite,  then 

lim  2  aK*"  =  I- 

x^\   1 

It  is  easy  to  verify  that   lim  ij,^*  =  lim  6-J^*  if  \  =  1  or  2  :  for 

higher  values  of  X  this  relation  certainly  holds  if  both  limits  exist, 
as  follows  from  Theorems  1  and  4. 

In  Part  II,  I  propose  to  extend  Theorem  3  to  certain  other 
mean  values;  and  Part  III  contains  some  general  remarks  about 
the  converse  of  Abel's  theorem. 

Part  I. 

].  In  the  researches  which  follow  I  have  to  make  use  of  the 
following  theorems. 

n 

Theorem  5.     //  lim  2  a«  =  lim  s,,.  =  I    (finite), 
and  Uk  is  positive  and 

n, 

lim  2  6k  =  lim  t^  =  ^  , 

1  " 
then  lim  ~^b^s^=  lim  s„  =  /     (-1). 

This  theorem  is  due  to  Stolz*. 

Theorem  6.  Suppose  that  b^  is  positive  and  26«  divergent; 
a)id  let  D  be  the  region  defined  by 

p<2cosi|^     {\ylr\^\lr^<^'Tr), 

where  i  —  x  =  pe'''^.     Further  suppose  that 

tb,\x''\/\^b^x''\<G, 

where  (f  is  a  finite  constant,  for  all  values  of  x  inside  the  region  D. 

*  Bromwich,  Injinitc  series,  p.  378. 

10—2 


132 


iJr  Kienast,  Extensions  of 


Finally  suppose  that  a^lbn  tends   to   tlie   lint  it  I  luhen  n  tends   to 
infinity.     Then 

lim  (S  a^afjt  b^,/f)  =  I (o), 

tuhen  X  approaches  1  along  any  path  inside  D. 

This  theorem  is  due  to  Pringsheim*.  It  is  to  be  supposed 
throughout  this  paper  that,  when  x  tends  to  1,  its  approach  to  1 
is  along  some  path  inside  B. 

Theorem  7.     If  the  radius  of  convergence  of  F  {x)  = '!E  a^x" 

is  r,  then 

lim  anX'"'  =0     (\x\<  r). 

If  the  radius  of  convergence  of  Q(x)  =  Xa^x"  is  unity,  it  will 

I 
remain  unchanged  if  Q(x)  be  transformed  in  any  of  the  following 
ways : 

(i)     by  suppressing  a  limited  number  of  terms, 

(ii)    by  multiplying  by  ./;°",  a  being  an  integer, 

(iii)   by  multiplying  by  ^  _  ^ ,  =  ^  «'\ 

(iv)    by  integrating  term  by  term, 

(v)    by  differentiating  a  limited  number  of  times. 

Using  in  succession  one  or  other  of  these  operations,  there 
result  the  following  power-series,  all  with  radius  of  convergence 
unity : 

xF'  (x)  —  S  Ka^x", 


\(x)=   —A^F'{x)]  =  lrl'>x, 

X  —  X  I 


F,(xy=    ^ 


^  1  ~l     '^     ) 

-  Pj  (x)  dx  \  =X  r,^  x", 


1  —  X 


P.^  (x)  dx 


Pg  (x)  dx 


=  ir^^x'^\ 


*  4Qta  Mathematica,  vol.  28,  p.  7. 


Abel's  Theorem  and  its  converses  133 

Thus  the  series    S  r*^' ,    ,.r'^"^^~"  converges  if  Lt  I  <  1 ;  the  same 
is  the  case  with  ^^ 

or  i     r(\+;\«,-'^^^-l 

J)  +  A  — 1 

Differentiating  the  last  series  (X  — 2)  times,  we  obtain 

SO^  +  1  )0;  +  2) ...  (^9  +  X  -  2)7-<^+;':;,a''^ ; 
which  gives 

00 

Theorem  8.     //  the  radius  of  convergence  of  P  (a-)  =  2  a^.r"  is 

1 
unity,  then  for  every  \x\<\ 

r  (A  +  k)       n+A-2        r. 

hm  ?■   , ,   ,  ./•  =  0, 

/l+A  — 1  ' 

hm  -  r        ic   =  0, 

lim  (>i  +  1 )  (7?  +  2)  . . .  (n  +  \  -  2)  r^+^^l^  «^"  =  0. 

2.     The    demonstration    of    Theorem    8    depends    on    certain 
identities.     The  formula 

'  n    " 

leads,  by  successive  summation,  to  the  series  of  equations 

(6). 


(2)  (1)         1      (2) 

n  I)  j^      /( 


^(A)  _    _(A-1I   _   J-   ^.(A) 


If  lim  s     =1  exists,  then,  bv  Theorem  5,  lira  s .        also  exists 

and  is  equal  to  /,  and  therefore  one  of  these  identities  gives 

r       1     (A+i)       r\ 
Inn  -  ?'  =  0. 

Theorem  9.     If  lim  .s^^'  =  I  exists  and  is  finite,  then 

n-*-x 

r  1        (A  +  l(  n. 

lim      7"  =  0. 

7l-»-X    " 

Thus  the  second  condition  of  Theorem  3  is  necessaiy. 


134 


Dr  Kievast,  E.vtensioiis  of 


3.     I  proceed  to  prove  some  other  identities.     We  have 


,(1) 


i  KQ 

K 
1 


a«  =  -  r     -/ 


U<i>_-,.(i'   ! 


,(^) 


,(A-|-1) 


1     K     " 


M) 


Jl  +  1  M 


.(7); 


and  by  successive  substitution  we  find 

1 

n  '  " 

(3)  Q,„.,(3)       ,     O/.,         1\^(3)        /„,        0\  ,,(3) 


"  '    -  n—l>  n  +  l  n 


+  /2)     +1|,.C2)_,,(2)    } 


=  (n  +  1)  C  -  3m'- ,  +  3  (.^  -  1)  r^'  -  {n  -  2) ,-- ^ 

fl  '■   n  n-l> 

Writing 


, .  (n  +  X  -  yu,  -  2)  r 


(\+k)  ^   ^ 

M+A  — M  — 1  K.K.m 


.(8). 


we  can  easily  verify  that 

6r,        =tT>.i       — (X  —  l)0.     ,,      

\,K,n.  \  +  l,K,n        ^  ■'      A,K+l,w 

Moreover 

Developing  a^  in  this  way  we  obtain,  after  a  finite  number  p 
of  steps,  the  formula 


cin=     S 


A+K  =  P 


A,  K      A,  K,  n    \^ 


iUp)_, 


.(p) 


.(10). 


The  upper  index  of  all  the  7''s  is  the  same  throughout  this  expres- 
sion. For  the  present  purpose  it  is  not  necessary  to  determine  the 
coefiicients  c^  ^,  which  are  integers. 

In  consequence  of  the  definitions  of  s      and  r      we  have 

7>)  =  0     (n>vi). 


Abel's  Theorem  and  'its  converses 


135 


But  it  is  not  difficult  to  see  that  the  recurrence  formulae  (8)  and 
(9)  still  hold,  if  the  number  p  of  steps  exceeds  the  index  n.  It  is 
only  necessary  to  put  ?-^"  =  0  whenever  n  >  m.  The  form  of  the 
relations  (7),  viz. 

is  the  cause  why  the  coefficients  of  the  remaining  terms  are  not 
influenced  by  the  fact  that  some  terms  disappear.     Thus 


'I''  A,  K  A,  K,  n 

1  A+K=p  \n=\ 


+  S  -a;^{7>'_7»  I  (11). 

n  '    »  n-V  V        ^ 

n=p  "• 

To  evaluate  the  first  of  these  sums  we  have 

(A+k) 


(1  -  xf  s  (« + 1 ) . . .  {n  +  X  -  2)  r;;;;:,  ^" 

n  =  \ 

in 


,»H+1' 


A,  (C,  7J 


\,K,Vl  +  V  ' 


say.     Each  of  the  i\(\  +  l)  terms  contained  in  the  second  sum 
has  the  form 

K{p  +  i)(p  +  2)...{p  +  x-2)rl^;;!y+', 

{fi=  V,  V  +  1,  ...\;    V  =  1,  2,  ...X;   p  =  m  —  v  +  1). 

Therefore,   by  Theorem   8,  we   have,  for  every  j  x  \  <  1   and   any 
finite  X, 


limS  G,^^^^.r"^(l-a^^{n  +  l){n+2)...{n+X-2)rl^^;;^^x'\ 


«4-*-»    1  1 

The  second  sum  in  (11)  gives 


'"-     1  .      N  ,      \  U-1  -\ 

vif,.(p)_^,.W  }./=  S  r^' 


(p) 


n  +  1 


;i         -'^  ^  (p)    m 

m   «' 


=  {\-x)     S     -   -  7>'  A'"  +    S        ~         ,  7>*  X''  +  i  7>^ .^'" 

p    II  +  \    "  p    II  (v  +  1)    "  m   '« 

(12); 


and   again,  by  Theorem   8,  we   have,  for  every   ,  x  j  <  1   and  any 
finite  p. 


lim  %    x'  {r^'^  -  9>\}  =  (1  ^  X)  2  — ^  7>\*'"  +  ^—^ 
m-»  p  *''          "  ""'  p»  +  l    "  ,n(n- 


+  1)    " 


,.(p)  _.,."_ 


136  Dr  Kienast,  Extensions  of 

Thus  we  have  established 

Theorem  lO.     //'  l.a^x^  has  unity  as  radius  of  convergence, 
then 


S  ancc"  =    t    c. 

1  A+K=p  ' 


00 


M  =  l 


.(f^)  ^,"  4.  V  1  ,,(P)  ,..« 


(13) 

4.  Equation  (13)  has  now  to  be  considered  when  .r->l.  To 
the  first  terms  on  the  right-hand  side  we  apply  Theorem  6,  which 
gives 

00 

2(n  +  l)...(n  +  x-2)r;;)^_^^-'^  ^» 

hm . =  lim  -Jl±h=l_ , 

T^^— jyi  2  (/i  +  1)  .  .  .  (h +X- 1)^'* 

Again,  by  Theorem  6, 

CO  1  -. 

lim  (1  -  ^;)  2  -4t  ^'I^^'"  =  lim  -  r^"* : 
and  finally  Theorem  8  gives 

Theorem  11.  If  ta^x"  has  unity  as  radius  of  convergence, 
and  if 

\im~rl^^  =  0, 

J  »  00  1 

then  hm  2  a^  x"  =  lim  2 - j-^"'  x" 

x^l   1  0,-^!   p  «(/?+!)     " 

5.     Furthermore,  equations  (7)  and  (12)  lead  to 

«''  1        ,  ,  ,  ,  m-l  n 

'"•"1       1  m—l  1  1 

P    w  +  1    "  p    7z(w+l)    ■"   ■     ^m    ^'^  •''   • 

Putting  X  =  1,  it  follows  that 

-2  ^ 7^^'"^=  2  L^,>+i»  ,   1  ,.(p+i) 


Abel's  Theorem  and  its  converses 


1:17 


Hence 

Theorem  12.      //'  Xok-"''"  Iki-'^  <inity  as  f(((Iiiis  of  convergence, 
and  if 

1 


then 


.(p) 


'^^        1 


r "  =  lini    2, 


,(p+i) 


I 


Another    identity  is   acquired    by   developing  s\f  (;?  =  p  +  l^ 
/)  +  2, ...)  in  the  form 


P  +  1 


P+2 


>)_JP)     1 


j(P-l) 


pVi 


p+2 


1  ^(p-1) 


1 V.*"-^' 


V/i(?W  —  1) 


I 


"-1  1  (   \ 


VI  (m  +  1) 


(p) 


til  en 


Theorem  13.     If  .s"'   and  r  '''  are  defined  as  in  (2)  and  (3) 


S^'^=     S 


.(P) 


If  lim  ,9''^*  =  I  exists  and  is  finite,  then,  by  Theorem  5, 

II  ^x 

r         ('^+1)       7. 
hm  s         = I : 

II 

ll->-orj 

and  by  Theorem  13 

S  1 


.(A+i) 


=  /. 


.(^+1) 


Therefore  by  Abel's  theorem 

TV  -'-  (A  +  1)      III  V  ■'- 

inn  2.     -  "Tvr         ,r    —  z  — 7 Ts 

x^i  A+i  ?H  (?/i  +  1 )    '"  A+i  m  (m  +  \)    '" 

On   the  same  assumption,  Theorem  9  gives    lim  -■^•^^"^  ■  =0; 
and  therefore  Theorem  1 1  gives 


=  /. 


lim  2  n.^x"  —  lim  2 


^,(A+l)     « 


1X+1«("   +   1)      " 


"  7 


1^8  Dr  Kienast,  Extensions  of 

Thus  we  obtain 

Theorem  4.     If  Km  s\';^  =  I  exists  and  is  finite,  th 


en 


Mni^a^x"  =  I. 

X-»-l    1 


The  first  condition  of  Theorem  3  is  therefore  necessary. 

6.  To  demonstrate  the  rest  of  the  assertion  in  Theorem  8,  it 
follows  from  the  hypothesis  lim  \/^^'^  =  0  that  Theorem  11  is 
applicable.     Thus  the  assumptions  are  transformed  into 


lim  2  -  ^rl^+'\-=.i 
lim-7^;^+^'=0. 

71-*- 00  ^i 

From  this  last  equation  follows 


I     ''  1  <>.^^^  ..         1 


.(A+l) 


lim:^S  -^r(^+^'=lim^^^  =  0 

,,-^00  nx+lK  +  l      «  ,,^^  71  +  1 

Hence  Theorem  2  can  be  applied  to  the  series  2 - x'' • 

11  ,     •       •  n{n  +  \)      ' 

and  the  conclusion  is  that 

lim   t  -r~~^,  =  I. 
Theorems  12  and  13  now  yield 

lim  4^^=/, 

with  which  the  proof  of  Theorem  3  is  completed. 

7.  The  foregoing  deductions  are  valid  for  X  =  l,  2,....  For 
X  =  0  they  still  hold,  except  those  in  §  6.  This  case  requires  the 
proof  of  the  following  special  case  of  theorem  2 : 


Theorem  14.     // 


limS^-i--r^V  =  /, 
^^1  1  n(n  +  1)    « 

lim  ~r^'^  =  0 


I 


Abel's  Theorem  and  its  converses 


180 


then 


r'''=l. 


1  n(n+l) 

This  proof  is  actually  given  by  Mr  Tanber,  and  is  therefore 
the  basis  of  the  theorems  of  this  paper. 


Part  II. 

8.     Let  6k  denote  the  terms  of  an  infinite  sequence  of  positive 
real  numbers,  which  have  the  properties 


(1)  lim  S  b^  =  lim  tn  =  oo 


.(14), 


(2) 


n  1 


tends  to  a  limit  or  oscillates  between  finite  limits.     Then 
Theorem  16.     The  two  conditions 

QO 

lim  S  «'««"'  =  /      (finite), 

x^l  1 

1  » 
hm  --l^AaA  =  0, 

n-*-cc  I'll   1 

are  each  necessary  for  the  convergence  of  S  a«,  i.e.  for  the  existence  of 

II 
lim  %aK=  I : 

and,  taken  together,  they  are  sufficient. 

Abel's   theorem   states   that    the   first   of  these   conditions   is 
necessary. 

If  lim  Sn  =  I,  then  lim  a„  =  0,  and  by  Theorem  -5 


1   " 

lim  -  S  6a.?a-i  =  lim 


ii^X     ^'11 


1    n  1    n 

--2&ASA--S6AfO 

'«  1  f»i.  1 


/. 


The  identity 


1  "  1  " 

f  ft  2  f«    1 


n(^w  gives,  as  a  consequence  of  lim  ,<?„  =  /, 


1 
Therefore  the  second  condition  is  necessary  too. 


nm      ^  uxu\ 


^^^  Dr  Kienast,  Extensions  of 

9.     To  prove  the  converse,  we  require  two  identities.     If 

J^  hi 


n 


we  have 

X  a.af  =f  ^  +  2  -  |^«  -  p,_,]  x^ 


/ — 

1        ''K  +  l 


'«+r 


^« 


a' 


w-1 

1       ^K+l  1  t^  ^K  +  i 

Putting  a;  =  1,  this  gives  the  identity 


.(15). 


_\^t<±i-t^  Pk    .  p 


L 


t       ^f 

''K+i        hi 


•(16). 


If  we  suppose  lim  ^  =  0,  it  follows  that 

lim^.r'»=0 

for  every  I  .t  |  <  1 ;  and,  by  Theorem  6, 

lim(l-^)i^,,-  =  0. 

Now  passing  in  (15)   to  the    limit    (first   n -^  oo    and    then 
a?— »  1  we  find  that  if  lim  -^'=0,  then 

lim  2  o„ a;"  =  lim  2  ""^^ -^JL.^"  n^j) 

Theorem  15  starts  from  the  assumptions 
lim  ^  =  0, 

n-^-x   I'll 

CO 

lim  S  Ui^x"  =  I. 

x^\   1 

The  first  assumption  shows  that  (17)  is  available  ;  and  this  equation 
gives,  with  the  second  assumption, 

hm  z       ~  -i-^ a*  =  /. 

x-^\  1         t.        L ,, 


Abels  Theorem  and  its  converses  141 

NoAv  Theorem  2  can  be  applied  to  the  series  S  --^ — -  j-^  m^", 


provided  that 


hiu  -1,K  — , —    J      =  0    (lo). 


Assuming    for    a    moment    that    this    condition    is    satisfied, 
Theorem  2  leads  to 

lim  ^i<±}-zi<l^  =  i. 

Il^-X      I  ix  *K+l 

and  (16)  gives  finally 

lim  Sn  =  I, 

proving  the  theorem,  which  is  the  analogue  to  Theorem  2. 

Condition  (18)  depends  on  the  6's  as  well  as  on  the  as;  but 
since  lim  J-^  =  0.  it  will  certainly  be  fulfilled  when 

-Zk -— 

n  I  Ik 

tends  to  a  limit  or  oscillates  finitely.     For,  e  being  given,  we  can 
choose  K  so  that 

I  ^      ^A+,  -t^  Pk     ^1    "^^  ^  tK+i  -tK  pK      ,   e  V  ^  ^A+i  -  tK 

—  ^  A. <    -      Zi     /^ 7 i —  A,  -  . 

n  1  Ik        t\+i        n      \  Ik        t^+i        "  <  t^\ 

We  may  suppose,  for  example,  that 

tn  =  ri'';    \ogn;    log  log  7i;... 

10.     Adding  to  the  notations  used  hitherto 

1^7  (1) 

/      .1       A     A-l  71    ' 

^lbj^,  =  sf, 

'"n  3 

S  Ox  ^ =  CJn  , 

2  ^A-l 

and  restricting  the  choice  of  the  numbers  b^  not  only  as  done  in  8, 
but  further  by  supposing  that  the  two  limits 

lin,      L    ^1 ,       lim  h+ip^h+^  ^1   (19) 


142  Br  Kienufit,  Eidenaioiiii  of 

shall  exist,  or  at  any  rate  that  the  functions  under  the  limit  sign 
shall  oscillate  finitely,  I  procffcd  to  prove 

Theorem  16.      TJie  two  cunditions 

CO 

lim  2  ««*■"  =  I     (Jinite), 

x^l   1 

M-S-OG    hi    2  fA— 1 

are  each  necessary  for  the  existence  of  the  limit 

lim  s^'^  =  l: 

n 

mid,  taken  together,  they  are  sufficient. 

It  is  not  possible  to  demonstrate  this  theorem  for  every  set  of 
numbers  6^.     The  following  example  shows  this. 

Mr  Riesz  has  pointed  out*  that 

1       'i,  1 

hm  ^  -  Sk 

WH-Qo  log  n  1  K 

exists  and  is  finite  in  the  case  of 


However,  Abel's  limit 


lim  2  Kr''^~'^^x'^ 

x^\  1 


does  not  exist,  as  the  function  behaves  like 

r(«)(iogi)"' 

when  *■— »1. 

].l.     The  demonstration  depends  on  some  identities  analogous 
to  those  employed  in  the  case  of  the  arithmetic  means,  viz. 


n 

^n 

- 

1 

Pi 

! 

(2) 

H 

11 

- 

1 

tn 

n 
t 

2 

h. 

tK- 

*  See  G.  H.  Hardy, '  Slowly  oscillating  series',  Proc.  London  Math.  Soc,  ser.  2, 
vol.  8  (1910),  p.  310. 


Abel's  Theui'em  cuid  its  converses 


143 


which  scries  of  relations  might  be  continued.  They  show  (in  con- 
junction with  Theorem  5)  that  lim  s'  =  /  whenever  lim  s  —  /, 
from  which  we  deduce 


Theorem  17.     //"  lim  s|J'  =  I  exists  <incl  is  finite,  then 

lim^ii,^  =  0. 
Thus  the  second  condition  of  Theorem  16  is  necessary. 


12.     We  have  also 


7     Pn—1 

'^n—i 


and  thus 


V    ,,      ...K    _    ^    'h  +  ^  'i"      ,.K    _    ^    ^«  +  l    '/«    "    ^A-l    „.K 

1  1  Ok+1  2       fx  f^K 


II -\ 

In. 
1       i«- 


=  (1-^) 


V^!^±LZLil,,,K_i^ 


^K  +  l  ^«.+  l 


^,  #«  -  ^«_i  ^«  -  (y«_i 

+  Z         ^ — T-?—  w'' 

■1       t^  b^ 


+ 


'Jn+i'^  ^n-^ 


^n+\  bn+i 


[  1       K+2  I  t>«+l  &«+2  ^J 

+    1 h ^  ^ i  h     *  • 

"n+1  "n+i  2  f/f  Ox 


Now  the  series 


has  a  radius  of  convergence  at  least  as  great  as  1,  since  lim  -=^  =  0 

f?t— 1 

and =■  -^  tends  to  a  limit  or  oscillates  finitely.     Thus 

n-1    ba  •' 

lira  p  X''  =  0 


144  Dr  Kieiiad,  E.dennwns  of 

for  every    x  <  1 ,  arul  therefore 

i  a.,-  =  {i-xfi  p"  ^«  +  (1  -  ^)  V  ^±LZL^J  ^  ^ 

Taking  account  of  the  conditions  (19),  it  follows  from  Theorem  6 
that 

lim(l  -xf%p'a:-  =  0, 

x^-\  1   Ok +2 

and  Iini  (1  -  x)t  ^tir_^«+>  f+1 ,..  =  0, 

X^\  1  Ok-I-1  0^+2 

SO  that  lim  %  a^x"  =  lim  S  ^"^^  ~  ^^  a;'^^^    (20). 

13.     Lastly  we  have  the  identity 


14     If  lim  6-|/^  =  ^  exists  and  is  finite,  then,  by  (21),  i  ^"^'  ""^^^ 


1  ^K  +  l 

converges  to  the  sum  I.     Therefore  by  Abel's  theorem 


and  since  (Theorem  17)   lim  ^  =  0,  equation  (20)  is  valid,  and 
thus 


lim  1a^x''  =  l. 

x^l  1 


We  have  therefore 


Theorem  18.     Let  the  coefficients  h^  be  chosen  so  as  to  satisfy 
the  conditions  (19).     Then,  if  lim  6'J/*  =  I  exists  and  is  finite. 


lira  2  a^x"  =  I. 

x^\   1 


The  hrst  condition  of  Theorem  16  is  consequently  necessary. 


Abel's  Theorem  and  its  converses  145 

15.     The  proof  of  the  converse  begins  with  equation  (20),  which 
is  valid  since  lini  ~  =  0.     Therefore 

X^-l    1  f/C  +  1 

This  is  equivalent  to  the  first  condition  of  Theorem  15.     But  the 
second  is  satisfied  too,  viz. 

lini  yit.  ^A^i^i  ^  n„,  1  [-^^^^  -())+...+  (fy,  -  r/„_,)] 

«-*-x  hi  2  tie  M-*-c»  f)i 

=  lim  '^  =  0. 
Thus  lim  S'^'^^'~'^''  =  ^. 


and,  by  equation  (21), 


lim  s[y  =  I, 


which  completes  the  demonstration. 

The  conditions  (14)  and  (19)  imposed  on  the  numbers  b^  are 
not  necessary  but  only  sufficient.  The  conditions  necessary  and 
sufficient  would  depend  also  on  the  coefficients  ««  of  the  power 
series  considered,  so  that  for  a  given  series  'S^a^x"  a  given  set  6^ 
)nay  be  admitted  which  must  be  excluded  for  other  series  ^CkX". 


Part  III. 

16.  Theorem  2  is  in  a  sense  a  perfect  converse  of  Abel's 
theorem,  from  which  all  these  researches  originated. 

Series  for  which  Abel's  limit  exists  may  be  divided  into  two 
classes,  those  which  are  convergent  and  those  which  are  divergent, 
series  for  which  the  limit  does  not  exist  being  excluded. 
Theorem  2  shows  that  the  first  class  consists  of  those,  and  those 
only,  which  satisfy  the  condition 

lim  -i«ct,  =  0    (22). 

The  second  class  consists  of  those,  and  those  only,  which  do  not 
satisfy  the  condition. 

The  condition  (22)  is  satisfied,  in  particular,  if 

lim  nau  =  0  (23) 

VOL.   XIX.   PAliT  IV.  11 


146  Dr  Kienast,  Extensions  of 

But  this  condition,  unlike  (22),  is  not  a  necessary  condition  for 
convergence. 

Recent  investigators  have  generalised  the  condition  (23)  in  a 
different  manner.    Thus  Mr  J.  E.  Littlewood  proved*  the  theorem  : 

"  2  a^  IS  convergent,  provided  lim  2  a^x^"  =  A  and  1  na„  \  <  K." 

1  x^l   1 

And  still  more  recently  Mr  G.  H.  Hardy  and  Mr  J.  E.  Littlewood f 
proved 

Theorem  19.     If  lim  2  a,,x'^  =  A ,  and  a„  >--K,  then  2 a„ 
x^i  n 

converges  to  the  sum  A. 

But  however  interesting  in  themselves  these  two  theorems  and 
their  proofs  may  be  they  are  less  perfect  than  Theorem  2.  For 
the  conditions  j  na,,  j  <  K  and  na,,  >  -  ^  are  neither  necessary  for 
convergence  nor  is  either,  together  with  \imta,x''  =  A,  necessary, 

nor  do  they  characterise  the  non-converging  series  for  which  Abel's 
limit  exists.  Their  interest  is  in  fact  of  a  quite  different  character 
from  that  of  Theorem  2. 

It  is  not  difficult  to  state  similar  theorems  which  are  open  to 
the  same  objection  but  which  give  information  in  cases  where  the 
last  two  theorems  fail. 

17.  The  terms  a^  of  any  sequence  can  be  written  in  the  form 
««  =  ^ »  where  t^  is  subject  to  the  same  conditions  as  in  Theorem  15. 
This  theorem  then  shows  that 

CO   ^  00  ■  T 

"  ^  -^  is  convergent,  provided  lim  2  ^-  a;"  =  ^  and  lim  -2c  =  0  " 

1  ''''  x^l  1  ^K  ,i-*oo  tn  1      " 

Now  the  second  condition  is  certainly  satisfied  if  lim  2  c«  tends 

to  a  limit  or  oscillates  finitely.  The  only  limitation'thus  imposed 
upon  the  order  of  magnitude  of  a«  is  that  \c^\<K,  i.e.  that  the 

order  of  /c  |  a«  |  does  not  exceed  that  of  ^  .    Instead  of  the  condition 

T    */'  ,^-  ^ittlewood,  'The  converse  of  Abel's  Theorem  on  power-series',  Proc 
London  Math.  Sac,  ser.  2,  vol.  9  (1911),  p.  438.  .  -^  '<^c. 

_t  G.H^ Hardy  and  J.  E.  Littlewood,  '  Tauberian  theorems  concerning,  power- 
series  and  Dirichlet  s  series  whose  coeflicients  are  ijositive ',  Proc.  London  Math    '^nr 
ser.  2,  vol.  13  (1914)    p.  188.     See  also  E.   LanJau,  I)^.^./..,;"^  ^^'^^^^ 
ciniger  neuerer  ErgebniHse  der  Funktionentheorie.  (Berlin,  1916)   pn   45  etsea  ■  ihl 
actual  theorem  is  stated  in  §  9  and  finally  proved  in  §  10  (Die  HardyiLittlewoodsche 
Umkehrung  des  Abelschen  Stetigkeitssatzes). 


Abel's  Theorem  and  its  converses  147 


««K  >  —  A"  of  Theorem  19  we  have  \X  t^a,,  <  K,  a  condition  which 

I  1 
allows  Kti^  to  tend  to  infinity  in  either  direction. 

That  such  cases  exist,  in  which  S  ««  is  convergent,  is  shown  by 
the  fact  that 

t-—    (0<e<</)<27r-e) 

is  convergent  if  t^  is  any  function  of  k  which  tends  steadily  to 
infinity  with  k. 

18.  A  similar  result  can  be  obtained  from  another  theorem  of 
Messrs  Hardy  and  Littlewood,  viz. : 

Theorem  20.  If  f{x)  =  S  a^af  is  a  potver  series  with  positive 
coefficients,  luul  f{x)^  ^ as  x^\,  then 

n 

2  a«  ~  /I.  * 
1 

From  this  theorem  it  is  possible  to  deduce  Theorem  19  (see 
above)  of  the  same  authors. 

Now  the  hypothesis  is  equivalent  to 

CO 

lim  (1  —  x)  f{x)  =  lim  %  (a^  —  a^-i)  x"  =  1, 
and  the  conclusion  is 

lim  -^a^=  hm  - S  J 2 (a^  -  «a-i) h  =  1. 

M-*M  '>l    1  M-*.0O  ^    1      (  1  j 

Thus  Theorem  20  is  equivalent  to 

CO  n 

Theorem  21.     If  limXb^x"  =  1,  and  if  the  sums  «„  =  ^  6« 

x-*l  1  1 

are  alt  positive,  then 

1  'i 
lim  -  2  6'k  =  1. 

»-*-oo  11    1 

Here  again  is  a  condition  which,  in  case  the  series  converges, 
does  not  prevent  the  real  numbers  Kb^  from  tending  to  infinity  in 
both  directions. 

*  G.  H.  Hardy  and  J.  E.  Littlewood,  I.e.     See  also  E.  Landau,  I.e.,  §  9. 


11  —  2 


148  Mr  Hardy,  Sir  George  Stokes  and  the 


Sir  George  Stokes  and  the  concept  of  uniform,  convergence.  Bv 
G.  H.  Hardy,  M.A.,  Trinity  College. 

[Received  1  Jan.  1918.     Read  4  Feb.  1918.] 

1.  The  discovery  of  the  notion  of  uniform  convergence  is 
generally  and  rightly  attributed  to  Weierstrass,  Stokes,  and  Seidel. 
The  idea  is  present  implicitly  in  Abel's  proof  of  his  celebrated 
theorem  on  the  continuity  of  power  series ;  but  the  three  mathe- 
maticians mentioned  were  the  first  to  recognise  it  explicitly  and 
formulate  it  in  general  terms*.  Their  work  was  quite  independent, 
and  it  would  be  generally  agreed  that  the  debt  which  mathematics 
owes  to  each  of  them  is  in  no  way  diminished  by  any  anticipation 
on  the  part  of  the  others.  Each,  as  it  happens,  has  some  special 
claim  to  recognition.  Weierstrass's  discovery  was  the  earliest,  and 
he  alone  fully  realised  its  far-reaching  importance  as  one  of  the 
fundamental  ideas  of  analysis.  Stokes  has  the  actual  priority  of 
publication ;  and  Seidel's  work  is  but  a  year  later  and,  while 
narrower  in  its  scope  than  that  of  Stokes,  is  even  sharper  and 
clearer. 

My  object  in  writing  this  note  is  to  call  attention  to  and,  so 
far  as  I  can,  explain  tw^o  puzzling  features  in  the  justly  famous 
memoir-f-  in  which  Stokes  announces  his  discovery.  The  memoir 
is  remarkable  in  many  respects,  containing  a  general  discussion  of 
the  possible  modes  of  convergence,  both  of  series  and  of  integi'als, 
far  in  advance  of  the  current  ideas  of  the  time.  It  contains  also 
two  serious  mistakes,  mistakes  which  seem  at  first  sight  almost 
inexplicable  on  the  part  of  a  mathematician  of  so  much  originality 
and  penetration. 

The  first  mistake  is  one  of  omission.  It  does  not  seem  to  have 
occurred  to  Stokes  that  his  discovery  had  any  bearing  whatever  on 
the  question  of  term  by  term  integration  of  an  infinite  series.  The 
same  criticism,  it  is  true,  may  be  made  of  Seidel's  paper.  But 
Seidel  is  merely  silent  on  the  subject.  Stokes,  on  the  other  hand, 
quotes  the  false  theorem  that  a  convergent  series  may  always  be 
integrated  term  by  term,  and  refers,  apparently  with  approval,  to 
the  erroneous  proof  offered  by  Cauchy  and  Moignoj. 

Of  this  there  is,  I  think,  a  fairly  simple  and  indeed  a  double 

*  The  idea  was  rediscovered  by  Cauchy,  five  or  six  years  after  tlie  publication  of 
the  work  of  Stokes  and  Seidel.  See  Pringsheim,  '  Grundlageu  der  allgemeineu 
Funktionenlehre ',  Encyld.  der  Math.  Wiss.,  II  A  1,  §17,  p.  35. 

t  '  On  the  critical  values  of  the  sums  of  periodic  series',  Trans.  Canib.  Phil.  Soc, 
vol.  8,  1847,  pp.  533-583  (Mathematical  and  physical  papers,  vol.  1,  pp.  236-313). 

X  See  p.  2-42  of  Stokes's  memoir  (as  printed  in  the  collected  papers). 


concept  of  uniform  convergence  •   149 

explanation.  In  the  first  place  it  must  be  remembered  that  Stokes 
was  primarily  a  mathematical  physicist.  He  was  also  a  most  acute 
pure  mathematician ;  but  he  approached  pure  mathematics  in  the 
spirit  in  which  a  physicist  approaches  natural  phenomena,  not 
looking  for  difficulties,  but  trying  to  explain  those  which  forced 
themselves  upon  his  attention.  The  difficulties  connected  with 
continuity  and  discontinuity  are  of  this  character.  The  theorem 
that  a  convergent  series  of  continuous  functions  has  necessarily 
a  continuous  sum  is  one  whose  falsity  is  open  and  aggressive : 
examples  to  the  contrary  obtrude  themselves  on  analyst  and 
physicist  alike.  The  falsity  of  this  theorem  Stokes  therefore 
observed  and  corrected.  The  falsity  of  the  corresponding  theorem 
concerning  integration  lies  somewhat  deeper.  It  is  easy  enough, 
when  one's  attention  has  been  called  to  it,  to  see  that  the  proof 
of  Cauchy  and  Moigno  is  invalid.  But  there  are  no  particularly 
obvious  examples  to  the  contrary :  simple  and  natural  examples 
are  indeed  somewhat  difficult  to  construct*.  And  Stokes,  his 
suspicions  never  having  been  excited,  seems  to  have  accepted  the 
false  theorem  without  examination  or  reflection. 

This  is  half  the  explanation.  The  second  half,  I  think,  lies  in 
the  distinctions  between  different  modes  of  uniform  convergence 
which  I  shall  consider  in  a  moment. 

Stokes's  second  mistake  is  more  obvious  and  striking.  He 
proves,  quite  accurately,  that  uniform  convergence  implies  con- 
tinuity f.  He  then  enunciates  and  otfers  a  proof  ;J;  of  the  converse 
theorem,  which  is  false.  The  error  is  not  one  merely  of  haste  or 
inattention.  The  argument  is  as  explicit  and  as  clearly  stated  in 
one  case  as  in  the  other ;  and,  up  to  the  last  sentence,  it  is  perfectly 
correct.  He  proves  that  continuity  involves  something,  and  then 
states,  without  further  argument,  that  this  something  is  what  he 
has  just  defined  as  uniform  convergence.  It  is  merely  this  last 
statement  that  is  false. 

Stokes's  mistake  seems  at  first  sight  so  palpable  that  I  was  for 
some  time  quite  at  a  loss  to  imagine  how  he  could  have  made  it. 
A  closer  examination  of  his  memoir,  and  a  comparison  of  his  work 
with  other  work  of  a  very  much  later  date,  has  made  the  lapse  a 
good  deal  more  intelligible  to  me  ;  and  my  attempts  to  understand 
it  have  led  me  to  a  number  of  remarks  which,  although  they 
contain  very  little  that  is  really  novel,  are,  I  think,  of  some 
historical  and  intrinsic  interest. 

2.  There  are  no  less  than  seven  different  senses,  all  important, 
in  which  a  series  may  be  said  to  be  uniformly  convergent. 

*   See  Bromwich,  Infinite  sfrlea,  pp.  110-118;  Hardy, '  Notes  on  some  points  in 
the  integral  calculus',  XL,  Messentjer  of  Matlieniatica,  vol.  44,  1915,  pp.  145-149. 
t  p.  282.     I  use  '  uniform  '  instead  of  Stokes's  '  not  infinitely  slow  '. 
X  p.  283. 


150  Afr  Hardy,  Sir  George  Stokes  and  the 

I  shall  write  the  series  in  the  form 

00 

S  tin  (^f^) ; 

1 
and  I  shall  suppose,  for  simplicity,  that  every  term  of  the  series  is 
continuous,  and  the  series  convergent,  for  every  x  of  the  interval 
a^x  ^b.    I  shall  denote  the  sum  of  the  series  by  s (x) ;  and  I  shall 
write 

Sn  (OC)  =  II,  (x)  +  Uo  {x)+  ...■\-  iln  (.«),     S  (x)  =  Sn  (x)  +  r,,  (cc). 

The  fundamental  inequality  in  all  my  definitions  will  be  of  the  t3^pe 

\rn(a!)'\^e (A), 

I  shall  refer  to  this  inequality  simply  as  (A). 

When  we  define  uniform  convergence,  in  one  sense  or  another, 
we  have  to  choose  various  numbers  in  a  definite  logical  order,  those 
which  are  chosen  later  being,  in  general,  functions  of  those  which 
are  chosen  before.  I  shall  write  each  number  in  a  form  in  which 
all  the  arguments  of  which  it  is  a  function  appear  explicitly :  thus 
no  (^,  e)  is  a  function  of  ^  and  e,  Uo  (e)  one  of  e  alone. 

It  will  sometimes  happen  that  one  of  the  later  numbers  depends 
upon  several  earlier  numbers  already  connected  hy  functional  rela- 
tions, so  that  it  is  really  a  function  of  a  selection  of  these  numbers 
only.  Thus  h  may  have  been  determined  as  a  function  of  e ;  and 
??o  niay  have  to  be  determined  as  a  function  of  ^,  e,  and  h,  so  that 
it  is  in  reality  a  function  of  ^  and  e  only.  I  shall  express  this  by 
writing 

«o  =  ^2o(^,  e,  S)  =  no(|^,  e); 
and  I  shall  use  a  similar  notation  in  other  cases  of  the  same  kind. 
3.  The  first  three  senses  of  uniform  convergence  are  as  follows. 
A  1 :  Uniform  convergence  throughout  an  interval.  The 
series  is  said  to  he  uniformly  convergent  throughout  the  interval  (a,  b) 
if  to  every  positive  e  corresponds  a.n  no  (e)  such  that  (A)  is  true^for 
n ^ Wo (e)  and  a^x^^b. 

This  is  the  ordinary  or  '  classical ',  and  most  important,  sense, 
the  sense  in  which  uniform  convergence  is  defined  in  every  treatise 
on  the  theory  of  series. 

A  2 :  Uniform  convergence  in  the  neighbourhood  of  a 
point.  The  series  is  said  to  be  uniformly  convergent  in  the 
neighbourhood  of  the  point  ^  of  the  interval  (a,  b)  if  an  interval 
(f  —  8  (I),  ^  -\-B  (^))*  can  be  found  throughout  luhich  it  is  uniformly 
convergent ;  that  is  to  say  %f  a  positive  8{^)  exists  such  that  (A) 
is  true  for  every  positive  e,  for  n  ^  n^  (^,  S,  e)  =  ??o  (?>  e),  and  for 

*  A  trivial  change  is  of  course  required  in  the  definition  if  t  =  «  or  ^  =  b.     The 
same  point  naturally  arises  in  the  later  definitions. 


concept  of  mil  form  convergence  151 

A3:  Uniform  convergence  at  a  point.  The  series  is 
said  to  be  uniformly  convergent  at  the  point  x  =  f  {or  for  x  =  ^) 
if  to  every  positive  e  correspond  a  positive  S  (^,  e)  and  an 
»o(|,  €,  B)  =  nQ(^,  e)  such  that  (A)  is  true  for  n  ^n^{^,  e)  and  for 
^-S(^,e)^x^^+S(^,e). 

4.  Before  proceeding  further  it  will  be  well  to  make  a  few 
remarks  concerning  these  definitions  and  their  relations  to  one 
another. 

The  idea  of  uniform  convergence  in  the  neighbourhood  of  a 
particular  point  (Definition  A  2)  is  substantially  that  defined  by 
Seidel  in  1848*.  It  is  clear,  however,  that  definitions  A  1  and 
A  2  were  both  familiar  to  Weierstrass  as  early  as  1841  or  1842f. 
It  is  obvious  that  a  series  uniformly  convergent  throughout  an 
interval  is  uniformly  convergent  in  the  neighbourhood  of  every 
point  of  the  interval.  The  converse  theorem  is  important  and  by 
no  means  obvious,  and  was  first  proved  by  Weierstrass |  in  a  memoir 
published  in  1880.  This  theorem  would  now  be  proved  by  a 
simple  application  of  the  '  Heine-Borel  Theorem ',  and  is  a  par- 
ticular case  of  a  theorem  which  will  be  referred  to  in  a  moment. 

Definition  A3  appears  first,  in  the  form  in  which  I  state  it,  in 
a  paper  of  W.  H.  Young  published  in  1903§;  but  the  idea  is 
present  in  an  earlier  paper  of  Osgood ||.  The  essential  difference 
between  definitions  A  2  and  A  3  is  that  in  the  latter  S  is  chosen 
after  e  and  is  a  function  of  ^  and  e,  while  in  the  former  it  is  chosen 
before  e  and  is  a  function  of  f  alone.  In  each  case  n^  is  a  function 
of  two  independent  variables,  ^  and  e.  It  is  plain  that  uniform 
convergence  in  the  neighbourhood  of  ^  involves  uniform  conver- 
gence at  ^,  and  at  (and  indeed  in  the  neighbourhood  of)  all  points 
sufficiently  near  to  ^.  But  uniform  convergence  at  ^  does  not 
involve  uniform  convergence  in  the  neighbourhood  of  |. 

It  is  important,  however,  to  observe  that  uniforni  convergence 
at  every  point  of  an  interval  involves  uniform  convergence  throughout 
tJie  interval.     This  important  theorem  is  proved  very  simply  by 

*  '  Note  iiber  eine  Eigensehaft  der  Reihen,  welche  discontinuirliche  Functionen 
darstellen',  Munchener  Ahliandlungen,  vol.  7,  1848,  pp.  381-394.  This  memoir  has 
been  reprinted  in  Ostwald's  Klassiker  der  e.vakten  Wisscmchaften,  no.  IK!.  The 
reference  tliere  given  to  vol.  5,  1847,  is  incorrect. 

(■  For  detailed  references  bearing  on  this  and  similar  historical  points,  see 
Pringsheim's  article  already  qnoted. 

X  See  the  memoir  'Zur  Functionenlehre '  {Ahliandlungen  aus  der  Funktionen- 
lehre,  pp.  69-104  (pp.  71-72)). 

§  'On  non-uniform  convergence  and  term-by-term  integration  of  series',  Proc. 
London  Math.  Soc,  ser.  2,  vol.  1,  pp.  89-102. 

II  'Non-uniform  convergence  and  the  integration  of  series',  American  Journal  of 
Math.,  vol.  19,  1897,  pp.  155-190.  See  Prof.  Young's  remarks  on  this  point  at  the 
beginning  of  his  later  paper  '  On  uniform  and  non-uniform  convergence  of  a  series 
of  continuous  functions  and  the  distinction  of  right  and  left ',  Proc.  London  Math. 
Soc,  ser.  2,  vol.  6,  1907,  pp.  29-51. 


152  Mr  Hardy,  Sir  George  Stokes  and  the 

Young,  in  his  paper  already  quoted,  by  means  of  the  Heine-Borel 
Theorem  * ;  and  it  plainly  includes,  as  a  particular  case,  Weierstrass's 
theorem  referred  to  above. 

5.  It  seems  to  me  that  the  definition  given  by  Stokes  is  not 
any  one  of  A  1 ,  A  2,  A  3  ;  and  that,  if  we  are  to  understand  him 
rightly,  we  must  consider  another  parallel  group  of  definitions. 
These  definitions  differ  from  those  given  above  in  that  (A)  is 
supposed  to  be  satisfied,  not  for  all  sufficiently  large  values  of  n, 
but  only  for  an  infinity  o/ values. 

B  1 :  Quasi-uniform  convergence  throughout  an  interval. 
The  series  is  said  to  he  quasi-uniformly  convergent  tlvroughout  (a,  h) 
if  to  every  positive  e  and  every  N  corresponds  an  n^  (e,  N)  greater  than 
N  and  such  that  (A)  is  true  for  n  =  n^  (e,  N)  and  a^x^b. 

B  2 :  Quasi-uniform  convergence  in  the  neighbourhood 
of  a  point.  The  series  is  said  to  be  quasi-uniformly  convergent  in 
the  neighbourhood  of  f  if  an  interval  (^  — 8(f),  |  +  S(f))  can  be 
found  throughout  which  it  is  quasi-uniformly  convergent ;  i.e.,  if  a 
positive  8(f)  exists  such  that  (A)  is  true  for  every  positive  e,  every  N,  an 
«o  (f .  8,  e,  iV)  =  ?io  (f ,  e.  ^)  greater  than  N,  and  f  —  8  (f )  ^  .'c  ^  f  +  5  (f ). 

B3:  Quasi-uniform  convergence  at  a  point,  llie  series 
is  said  to  be  quasi-uniformly  convergent  for  iV  =  ^  if  to  every  positive 
€  and  every  N  correspond  a  positive  S  (f,  e,  N)  and  an 

no{^,e,8,N)  =  n,(^,e,N), 

greater  than  N,  such  that  (A)  is  true  for  n  =  ??o  (!>  f>  N')  and  for 

Definition  B  1  is  to  be  attributed  to  Dini  or  to  Darboux+. 
Another  form  of  it  has  been  given  by  Hobson|.  As  Arzela  and 
Hobson§  have  pointed  out,  a  series  is  quasi-uniformly  convergent 
throughout  an  interval  if,  and  only  if,  it  can  be  made  uniformly 
convergent  by  an  appropriate  bracketing  of  its  terms. 

Definition  B  2  is  for  us  at  the  moment  of  peculiar  interest, 
for  (as  I  shall  show  in  a  moment)  it  is  really  this  definition  that 
is  given  by  Stokes. 

.  Definition  B  3  is  also  of  great  interest,  both  in  itself  and  in 

*  Choose  €  and  determine  5  (^,  e)  and  n^  (|,  e),  as  in  definition  A3,  for  every  f  of 
the  interval.  Every  point  of  {a,  b)  is  included  in  an  interval  {^-d,  ^  +  o).  By  the 
Heine-Borel  Theorem,  every  point  of  (a,  b)  is  included  in  one  or  other  of  a  finite 
sub-set  of  these  intervals.  If  N  (e)  is  the  largest  of  the  Hq's  corresponding  to  each  of 
the  intervals  of  this  finite  sub-set,  then  (A)  is  true  for  n^N  and  a  ^  .r  ^  6. 

This  is  the  essence  of  the  proof,  though,  like  all  proofs  of  the  same  character,  it 
requires  a  somewhat  more  careful  statement  if  all  apj^earance  of  dfpendence  upon 
Zermelo's  AusicahUprinzip  is  to  be  avoided. 

t  See  Pringsheim,  I.  c. 

+  '  On  modes  of  convergence  of  an  infinite  series  of  functions  of  a  real  variable', 
Proc.  London  Math.  Sac,  ser.  2,  vol.  1,  1903,  pp.  373-387.  Hobson  (following  Dini) 
uses  the  expression  '  simply  uniformly'. 

§  L.  c,  p.  375. 


concept  of  uniform  convergence  153 

relation  to  Stokes's  memoir.  For  the  necessary  and  sufficient  con- 
dition that  s  (x)  should  he  continuous  for  x=^  is  that  the  series 
should  be  quasi-uniformly  convergent  for  x  —  ^.  This  theorem  is 
in  substance  due  to  Dini*.  I  give  the  proof,  as  it  is  essential  for 
the  criticism  of  Stokes's  memoir. 

(1)  The  condition  is  siificient.     For 

I  s  {x)  -  s  (I)  I  ^  {  Sn  {x)  -  Sn  (|) '  +  |  r„  {x)  j  +  |  r^  (f)  |. 

Choose  e,  N,  S  (^,  e,  N),  and  n  =  ??o  {^,  e,  N)  as  in  definition  B  3.  Then 
[  r„  {x)  I  <  e  for  ^—Z^x^^  +  h.  Now  that  n  is  fixed  we  can  choose 
Si  less  than  8  and  such  that  [  s,i  {x)  —  «» (^)  j  <  e  for  ^  —  Si  ^  a"  ^  ^  +  Sj. 
And  thus 

|s(.«)-s(f)i<3e 

for  ^  —  §1  ^ .«  ^  ^  +  Si ,  so  that  5  {x)  is  continuous  for  a?  =  f . 

It  is  plain  that  this  argument  proves,  a  fortiori,  that  A  2,  A  3, 
and  B  2  all  furnish  sufficient  conditions  for  continuity  at  a  point, 
and  A 1  and  B 1  sufficient  conditions  for  continuity  throughout  an 
interval. 

(2)  The  condition  is  necessary.     For 

I  rn  {x)  \^\S  {x)  -  S  (^)  I  +  !  Vn  (f )  |  +  !  S„  {x)  -  5„  (|)  |. 

Suppose  that  e  and  N  are  given.  Then  we  can  choose  S  (^,  e) 
so  that  \s{x)  —  s{^)\<e  for  f  —  S  ^  «  ^  ^  +  S,  and  n^  (|,  e,  iV )  so  that 
Vq  >  N  and  j  r^^  (|)  |  <  e.  And,  when  n^  has  thus  been  fixed,  we  can 
choose  S]  (^,  e,  n^)  =  Sj  (^,  e,  N)  so  that  Si  <  S  and  1 6'„^  {x)  —  Sn^  (f )  |  <  e 
for  I  —  Si  ^  .^■  ^  I  +  Si .  Thus  |  r,i  (^)  j  <  3e  for  n  =  no>  N  and 
^  —  Bi^X'^^  +  8i,  so  that  the  series  is  quasi-uniformly  convergent 
for  x=^. 

6.  If  a  series  is  uniformly  convergent  at  every  point  ^  of  an 
interval,  it  is  (as  we  saw  in  §  4)  uniformly  convergent  throughout 
the  interval :  definition  A  3  (and  a  fortiori  definition  A  2)  passes 
over,  in  virtue  of  the  Heine-Borel  Theorem,  into  definition  A  1. 
It  is  important  to  observe  that  this  relation  does  not  hold  between 
B  3  (or  B  2)  and  B  1  :  a  series  quasi-uniformly  convergent  at 
every  point  of  an  interval  (or  in  the  neighbourhood  of  every  such 
point)  is  not  necessarily  quasi-uniformly  convergent  throughout 
the  interval.  We  can  apply  the  Heine-Borel  Theorem  in  the 
manner  indicated  in  the  first  sentences  of  the  footnote  *  to  p.  152  ; 
but  the  last  stage  of  the  argument,  in  which  every  one  of  a  finite 
number  of  difterent  integers  is  replaced  by  the  largest  of  them, 
fails.  What  we  obtain  is  the  necessary  and  sufficient  condition  that 
s  {x)  shoidd  he  continuous  throughout  the  interval ;  and  this  is  not 

^'  Foiulaiiii')iti...,  p.  107  ((jerinan  translation,  GruiuUa(ii'ii...,p\).  143-145). 


154  Mr  Hardy,  Sir  Georr/e  Stokes  and  the 

the  condition  B  1   but  a  condition  first  foi-mulated  by  Arzela*, 


VIZ. 


C:  Quasi-uniform  convergence  by  intervals  {convergenza 
uniforme  a  tratti).  ^  The  series  is  said  to  he  quasi- uniformly  con- 
vergent by  intervals  if  to  every  positive  e  and  every  N  correspond  a 
division  of  (a,  h)  into  a  finite  number  v  (e,  N)  of  intervals  8,.  (e,  N), 
and  a  corresponding  number  of  numbers  n,(e,  Nj,  all  greater  than  A^, 
and  such  that  (A)  is  true  for  ??  =  7?,.(?-  =  1,  %  ...,v)  and  all  values 
of  X  which  belong  to  8,.. 

The  deduction  of  Arzela's  criterion  from  B  3,  in  the  manner 
sketched  above,  was  first  made  by  Hobsonf. 

There  is  one  further  point  which  seems  worth  noticing  here, 
although  it  is  not  directly  connected  with  Stokes's  memoir.  Dini  J 
proved  that  if  u^  (x)  ^  0  for  all  values  of  n  and  x,  and  s  (x)  is  con- 
tinuous throughout  {a,  b),  then  the  series  is  uniformly  convergent 
throughout  (a,  b).  This  theorem  is  now  almost  intuitive.  For  it 
is  obvious  that,  for  series  of  positive  terms,  quasi-uniform  conver- 
gence in  any  one  of  the  senses  B  1,  B  2,  or  B  3  involves  uniform 
convergence  in  the  corresponding  sense  A  1,  A  2,  or  A  3.  If  then 
s  {x)  is  continuous  throughout  (a,  b)  it  is  continuous  for  every  f  of 
(a,  b) ;  and  therefore  the  series  is  quasi- uniformly  convergent  for 
every  f ;  and  therefore  uniformly  convergent  for  every  |;  and 
therefore  uniformly  convergent  throughout  (a,  b). 

7.  Let  us  now  consider  Stokes's  definitions  and  proofs  in  the 
light  of  the  preceding  discussion. 

It  is  clear,  in  the  first  place,  that  Stokes  has  in  his  mind  some 
phenomenon  characteristic  of  a  small,  hit  fixed,  neighbourhood  of 
a  point. 

'  Let  u^  -{-U.+  ...  (66)',  he  says§, '  be  a  convergent  infinite  series 
havmg  U  for  its  sum.  Let  v,  + v,  ■]-...  (Q7)  be  another  infinite 
series  of  which  the  general  term  v.,,  is  a  function  of  the  positive 
variable  h  and  becomes  equal  to  Un  when  h  vanishes.  Suppose 
that  for  a  sufiiciently  small  value  of  h  and  all  inferior  values  the 
series  (67)  is  convergent,  and  has  V  for  its  sum.  It  might  at  first 
sight  be  supposed  that  the  limit  of  V  for  A=0  was  necessarily 
equal  to  U.     This  however  is  not  true.... 

'  Theorem.  The  limit  of  V  can  never  differ  from  U  unless 
the  convergency  of  the  series  (67)  becomes  infinitely  slow  when  h 
vanishes. 

*  '  Sulle  serie  di  funzioni',  Memorie  dl  Bologna,  ser.  5,  vol.  8,  1900  up  131-186 
701-744.  ' 

t  L.  c,  pp.  380-382. 

J  L.c.  (German  edition),  pp.  148-149.  See  also  Bromwich,  Infinite  series,  p  125 
(Ex.  6).  ■  '^' 

§  p.  279. 


concept  of  II inform  convergence  155 

'  The  convergency  of  the  series  is  here  said  to  become  infinitely 
slow  when,  if  n  be  the  number  of  terms  which  must  be  taken  in 
order  to  render  the  sum  of  the  neglected  series  numerically  less 
than  a  given  quantity  e,  which  may  be  as  small  as  we  please,  n 
increases  beyond  all  limit  as  h  decreases  beyond  all  limit. 

'Demonstration.  If  the  convergency  do  not  become  in- 
finitely slow  it  will  be  possible  to  find  a  number  n,  so  great  that 
for  the  value  of  h  tue  begin  with  and  for  all  inferior  values  greater 
than  zero  the  sum  of  the  neglected  terms  shall  be  numerically  less 
than  e....' 

Stokes's  words,  and  in  particular  those  which  I  have  italicised, 
seem  to  me  to  make  two  things  perfectly  clear. 

(1)  Stokes  is  considering  neither  a  property  of  an  interval 
(a,  b)  im  Grossen  (such  as  is  contemplated  in  A  1  or  B  1),  nor  a 
property  of  a  single  point  which  (as  in  A  3  or  B  3)  need  not  be 
shared  by  any  neighbouring  point,  but  a  property  of  an  interval 
im  Kleinen,  that  is  to  say  a  small  but  fixed  interval  chosen  to  in- 
clude a  particular  point.  His  definition  is  therefore  one  of  the 
type  of  A  2  or  B  2. 

Stokes's  failure  to  perceive  the  bearing  of  his  discovery  on 
problems  of  integration  is  made  much  more  natural  when  we 
realise  that  he  is  considering  throughout  a  neighbourhood  of  a 
point  and  not  an  interval  im  Grossen.  And  this  remark  applies 
to  Seidel  as  well. 

(2)  Stokes  is  considering  an  inequality  satisfied  for  a  special 
value  of  n,  or  at  most  an  infinite  sequence  of  values  of  oi,  and  not 
necessarily  for  all  values  of  n  from  a  certain  point  onwards.  In 
this  respect  there  is  a  quite  sharp  distinction  between  Stokes's 
work  and  Seidel's.  What  Stokes  defines  is  (to  use  the  language 
of  this  note)  a  mode  of  quasi-unifo7'ni  convergence  and  not  one  of 
strictly  uniform  convergence. 

It  seems  to  me,  then,  that  what  Stokes  defines  is  what  I  have 
called  quasi-uniform  convergence  in  the  neighbourhood  of  a,  point 
(B2). 

8.  If  we  adopt  this  view,  Stokes's  mistake  becomes  very  much 
more  intelligible.  He  proves,  quite  correctly,  that  uniform  con- 
vergence in  his  sense  implies  continuit}^ :  his  proof,  stated  quite 
formally  and  by  means  of  inequalities,  is  substantially  that  given 
in  1 5,  under  (1).     He  then  continues*  as  follows. 

'  Conversely,  if  (66)  is  convergent,  and  if  U=  Vof,  the  con- 
vergency of  the  series  (67)  cannot  become  infinitely  slow  when  h 

*  p.  282.     Tbe  italics  are  mine. 

t  Ffl  is  what  Stokes  calls  'the  value  of  V  for  h  =  0',  by  which  he  means,  of 
course,  its  limit  when  h  tends  to  0. 


156    Mr  Hardy,  Sir  George  Stokes  and  uniform,  convergence 

vanishes.     For  if  Un,  V^  represent  the  sums  of  the  terms  after 
the  nth  in  the  series  {QQ),  (67)  respectively,  we  have 

V^Vn  +  V,:,  U=U^+Un'; 
whence 

v,:=Y-u-{v,,-u,,)^-u,:. 

Now  V-U,  Yn-  Un  vanish  with  h,  and  Ua  vanishes  when  n 
becomes  infinite.  Hence  for  a  sufficiently  small  value  of  h  and 
all  inferior  values,  together  with  a  value  of  n  sufficiently  large  and 
independent  of  h,  the  value  of  F,/  may  be  made  numerically  less 
than  ^  any  given  quantity  e  however  small ;  and  therefore,  by 
definition,  the  convergency  of  the  series  (67)  does  not  become  in- 
finitely sloiv  when  h  vanishes.' 

Now  this  argument  is,  until  we  reach  the  last  sentence,  perfectly 
accurate,  and  indeed,  if  we  translate  it  into  inequalities,  substantially 
identical  with  that  given  in  §  5,  under  (2).  Stokes  proves,  in  fact, 
that  continuity  at  |  involves  quasi-uniform  convergence  at  |. 
Where  he  falls  into  error  is  simply  in  his  final  assertion  that  this 
property  is  that  which  he  has  previously  defined,  the  mistake  being 
due  to  a  failure  to  observe  that  his  intervals  of  values  of  h  depend 
upon  a  prior  choice  of  e.  In  a  word,  he  confuses,  momentarily, 
B  2  and  B  3.  The  ordinary  view  that  Stokes  defined  uniform 
convergence  in  the  same  sense  as  Weierstrass  compels  us  to  suppose 
that  he  confused  B  3  with  A  1 ,  or  at  any  rate  with  A  2  :  and  this 
is  hardly  credible. 

I  add  one  final  remark.  If  we  could  identify  Stokes's  idea  with 
B_3,  instead  of  with  B  2,  we  could  acquit  him  of  having  made  any 
mistake  at  all,  since  B  3  really  is  a  necessary  and  sufiicient  con- 
dition for  continuity.  We  could  then  regard  Stokes  as  having 
anticipated  Dini's  theorem.  This  view,  however,  does  not  seem  to 
me  to  be  tenable. 


• 


Mr  Lake,  Shell-deposits  fornied  by  the  flood  of  January/,  1918     157 


Shell-deposits  formed  by  the  flood  of  January,  1918.  B}-  Philip 
Lake,  M.A.,  St  John's  College. 

[Read  18  February  1918.] 

The  heavy  snow  of  the  third  week  in  January  1918  was  followed 
by  a  very  rapid  thaw  and  a  considerable  fall  of  rain,  and  the  Cam, 
in  consequence,  rose  to  an  exceptional  height.  In  the  neighbour- 
hood of  Cambridge  the  floods  were  the  most  extensive  of  recent 
years,  the  water  reaching  its  highest  level  on  Sunday,  Jan.  20. 

The  traces  of  the  flood  remained  visible  for  several  weeks,  its 
limits  being  marked  in  most  places  by  straws,  twigs,  silt,  etc.,  with 
a  sprinkling  of  land  and  fresh-water  shells.  But  below  the  town, 
near  the  railway-bridge,  the  shells  were  so  abundant  as  to  form  a 
remarkable  deposit,  which  seems  to  deserve  a  special  record.  It 
was  not  till  the  25th  Jan.  that  I  saw  it,  and  the  following  notes 
are  drawn  up  from  the  observations  made  on  that  day  and  on  two 
or  three  subsequent  visits. 

The  deposit  lay  partly  upon  the  tow-path  and  partly  in  the 
shallow  ditch  on  the  iimer  side  of  the  path,  and  it  extended  with 
little  interruption  from  the  immediate  neighbourhood  of  the  'Pike 
and  Eel '  to  a  point  about  850  yards  below  the  railway-bridge,  a 
total  distance  of  approximately  850  yards.  Occasional  patches 
occurred  still  farther  down,  and  scattered  shells  even  as  far  as 
Ditton  Corner.  Beyond  Ditton  the  tow-path  was  in  several  places 
covered  with  a  thick  layer  of  silt,  but  I  saw  no  more  shells  until 
within  sight  of  the  lock  at  Baitsbite. 

The  deposit  was  somewhat  irregular  and  it  was  difficult  to  form 
an  estimate  of  its  average  width,  but  this  can  hardly  have  been 
less  than  a  foot,  and  was  probably  much  more. 

Above  the  railway-bridge  the  shells  were  mixed  with  silt, 
especially  in  the  ditch  on  the  inner  side  of  the  path ;  but  even 
here  the  proportion  of  shells  was  large,  and  in  places  they  formed 
the  bulk  of  the  deposit.  Below  the  railway-bridge  the  deposit  was 
free  from  silt  and  consisted  entirely  of  shells.  In  the  shallow 
hollows  formed  by  the  irregularities  of  the  surface,  it  was  often  an 
inch  or  two  deep,  so  that  it  was  possible  to  scoop  up  the  shells  by 
the  handful.  Owing  to  its  colour  it  showed  conspicuously  as  light 
streaks  upon  the  slightly  darker  path. 

By  far  the  greater  part  of  the  deposit  consisted  of  Limnaea, 
L.  stagnalis  and  L.  peregra  being  the  most  abundant  species  ;  but 
other  fresh-water  shells  also  occurred  and  land-snails  were  by  no 


158  Mr  Lake,  Skell-depu'sits  foniied  bij  the 

means  rare.  Mr  C.  E.  Gray,  of  the  Sedgwick  Museum,  went  down 
shortly  after  my  first  visit,  and  in  a  very  short  time  obtained  most 
of  the  following  species,  but  a  few  names  have  been  added  to  the 
list  from  specimens  collected  subsequently : 

Sphaerium  corneum  (L.), 
Bithynia  tentaculata  (L.), 
Vivipara  contecta  (Millet), 
Valvata  piscinalis  (Miiller), 
Limnaea  stagnalis  (L.), 

„         peregra  (Mtiller), 

„         auricularia  (L.), 
Pkuiorhis  corneus  (L.), 

„  umbilicatus  Miiller, 

„  caiinatits  Miiller, 

„  vortex  (L.), 

„  contortus  (L.), 

Pliysa  fontinalis  (L.), 
Helix  nemoralis  L., 
Theba  cantiana  (Mont.), 
Hygromia  striolata  (Pfr.), 
Vitrea  draparnaldi  (Beck), 
„       cellaria  (Miiller). 

Even  now  the  list  is  probably  far  from  complete,  and  a  closer 
examination  would  no  doubt  reveal  the  presence  of  many  other 
forms. 

The  last  five  species  are  land-shells,  and,  with  the  exception  of 
Vitrea  cellaria,  they  occurred  in  Mr  Gray's  first  collection  and  were 
identified  by  Mr  Hugh  Watson.  Vitrea  draparnaldi  does  not 
appear  to  be  a  native  of  the  county,  but  is  found  in  and  near  green- 
houses ;  for  instance,  in  the  Botanical  Gardens.  In  Mr  Gray's  first 
collection,  which  was  made  below  the  railway-bridge,  it  was  repre- 
sented only  by  a  single  specimen,  which  we  supposed  to  have  come 
from  the  florist's  greenhouses  close  by.  But  at  a  later  date  he 
found  it  to  occur  abundantly  at  the  beginning  of  the  tow-j)ath, 
some  five  or  six  hundred  yards  above  the  greenhouses.  In  order 
to  make  sure  that  the  specimens  really  belong  to  this  species  they 
were  sent  to  Mr  Watson,  who  agreed  with  the  identification. 

Since  there  were  so  many  specimens  of  Vitrea  draparnaldi  at 
the  beginning  of  the  tow-path,  and  so  few  (at  least  comparatively) 
below  the  railway-bridge,  it  seems  clear  that  they  cannot  have  been 
carried  far,  for  otherwise  they  would  have  been  more  evenly  dis- 
tributed. It  is  most  probable  indeed  that  there  was  a  colony  of  this 
species  in  the  immediate  neighbourhood.  The  nearest  greenhouse 
that  I  have  been  able  to  find  above  the  locality  where  the  species 
was  so  abundant  is  five  or  six  hundred  yards  off,  and  stands  well 


I 


flood  (if  Jcumuvij,  1918  159 

away  from  the  river.  The  specimens  can  hardly  have  come  from 
there,  and  it  is  more  likely  that  the  colony  lived  out  of  doors  and 
nearer  to  the  river.  Nevertheless  its  progenitors  may  have  been 
'escapes'.  The  greenhouses  below  the  railway-bridge  have  now 
been  out  of  use  for  some  time,  and  the  snails  that  were  in  them 
must  have  been  forced  to  seek  new  quarters. 

Most  of  the  shells,  both  land  and  fresh-water,  were  perfect  or 
nearly  so,  and  all  of  them  were  empty.  Neither  Mr  Gray  nor  myself 
found  a  single  specimen  with  any  remains  of  its  former  inhabitant. 
The  greater  number  were  very  fresh  in  appearance,  but  some  of 
the  land-shells  had  evidently  been  exposed  to  the  weather  for  some 
time,  and  some  of  the  fresh-water  shells  had  lain  in  the  mud  long- 
enough  to  become  discoloured  or  incrusted  as  if  the  process  of 
fossilization  had  begun.  The  specimens  of  Vitrea  draparnaldi,  it 
may  be  noted,  were  all  fresh-looking. 

Apart  from  the  extent  of  the  shelly  deposit,  its  freedom  from 
silt  below  the  railway-bridge  was  perhaps  its  most  important  feature, 
for  it  shows  that  even  a  muddy  river  like  the  Cam  may  produce  a 
purely  calcareous  deposit. 

The  fact  that  the  shells  were  all  empty  indicates  that  those 
belonging  to  the  river  must  have  lain  in  its  bed  for  some  time;  and 
in  this  connection  an  observation  made  by  Mr  Gray  is  of  interest. 
Some  years  ago  at  Bottisham,  when  dredging  operations  were  going 
on,  he  noticed  that  the  mud  brought  up  by  the  dredger  was  full  of 
fresh-water  shells. 

During  floods  the  river  digs  up  its  bed  and,  as  on  the  occasion 
here  described,  it  may  deposit  the  shells  in  one  place  and  the  silt 
in  another.  In  the  case  of  an  artificially  controlled  stream  like  the 
Cam,  floods  are  comparatively  rare ;  but  in  an  unrestrained  river 
we  may  reasonably  expect  them  to  be  both  more  numerous  and 
more  extensive.  It  seems  quite  possible  therefore  that  neither  the 
clayey  fresh-water  limestones  of  the  Wealden  nor  the  purer  fresh- 
water limestones  of  the  Purbeck  series  required  lagunary  conditions 
for  their  formation. 


160  Mr  Matthui,  la  the  Madveporarian  Skeleton 


Is  the  Madveporarian  Skeleton  an  Extraprotoplasmic  Secretion 
of  the  Polyps  ?  By  G.  Matthai,  M.A.,  Emmanuel  College,  Cam- 
bridge.   (Communicated  by  Professor  Stanley  Gardiner.) 

[Read  18  February  1918.] 

In  1881  von  Heider  (5)  suggested  that  the  calcareous  skeleton 
of  the  Madreporaria  is  formed  by  the  deposition  of  carbonate  of  lime 
within  certain  specialised  ectodermal  cells  (calicoblasts*)  consti- 
tuting an  outer  layer,  and  repeated  this  conclusion  in  a  subsequent 
paper  (6).  In  1882  von  Koch  (8)  inferred  from  embryological  obser- 
vations that  the  skeleton  is  deposited  outside  the  living  tissues, 
i.e.  is  extraprotoplasmic  in  origin.  In  1896  Ogilvie  (9)  supported 
von  Heider's  view  and  argued  that,  by  repeated  calcification  of 
"cells"  of  the  calicoblastic  layer  of  ectoderm,  successive  strata  of 
calcareous  "  scales  "  are  formed,  and  slightly  modified  her  opinion 
in  1906  (10).  Fowler  (4)  had  previously  accepted  von  Koch's  view. 
In  1899  Bourne  (2),  from  his  studies  on  the  Anthozoan  skeleton, 
supported  von  Koch's  conclusions  and  entirely  disagreed  with 
von  Heider  and  Ogilvie.  He  further  held  that,  whilst  in  Heliopora 
and  the  Madreporaria  the  corallum  is  formed  outside  the  living 
calicoblastic  layer,  the  spicules  of  the  Alcyonaria  are  formed  within 
certain  ectodermal  cells  or  scleroblasts  which  either  remain  in  the 
ectoderm  or  wander  into  the  mesoglaea  (2,  p.  506).  Following 
von  Koch  and  Bourne,  it  is  noAv  generally  believed  that  the 
Madreporarian  skeleton  is  an  extraprotoplasmic  formation  and  that 
Alcyonarian  spicules  are  entoplastic  products. 

After  a  ground-down  section  of  an  Astrgeid  corallite  has  been 
slowly  decalcified  on  a  slide,  somewhat  homogeneous  organic 
remains  (distinguishable  from  algal  filaments  penetrating  the 
skeleton)  are  left  which  react  to  any  of  the  common  stains.  This 
is  clear  indication  that  the  calcareous  matter  has  been  deposited 
in  an  organic  matrix.  Bourne  regards  this  matrix  as  due  to  the 
"disintegration  of  calicoblasts"  (2,  pp.  520  and  521,  fig.  21), 
assuming  that  the  organic  basis  was  not  part  of  the  living  calico- 
blastic ectoderm.  His  view  is  that  carbonate  of  lime  is  secreted 
by  the  calicoblastic  layer  and  is  passed  through  its  outer  border 
(the  "  limiting  membrane  ")  into  the  decaying  part  outside,  exactly 
as  the  Alcyonarian  spicule  is  "  from  its  early  origin,  separated 
from  the  protoplasm  which  elaborated  the  material  necessary  for 
its  further  growth  by  a  layer  of  some  cuticular  material"  (2,  p.  537), 

*  Von  Heider's  original  rendering  of  this  word  is  chalicoblast,  of  which  the  first 
half,  I  am  informed,  is  derived  from  the  Greek  x'^^'li  which  in  Eomau  characters 
should  be  spelt  clialix.  Subsequently,  Fowler  changed  the  spelling  to  calycoblast, 
and  in  1888  both  this  author  and  Bourne  adopted  the  present  form  calicob/ast. 


an  Extraprotoplasmic  Secretion  of  the  Polyps!  161 

viz.,  the  spicule-sheath.  At  the  same  time,  Bourne  contends  that 
the  spicule  is  entoplastic  in  formation  whilst  the  Madreporarian 
coralkim  is  exoplastic.  To  be  consistent,  both  the  spicule  and  the 
corallum  would  have  to  be  regarded  as  formed  either  within  living 
protoplasm  or  outside  it,  but  spicules  could  not  be  viewed  as  intra- 
protoplasmic  products  whilst  assuming  the  extraprotoplasmic  origin 
of  the  corallum. 

Duerden  (3)  held  that  the  organic  basis  of  the  corallum  of 
Siderastrea  yadians  is  a  "  secretion  "  of  the  calicoblastic  layer  of 
ectoderm  to  which  it  is  closely  adherent  (pi.  8,  fig.  45)  and  is  "a 
homogeneous,  mesoglaea-like  matrix  within  which  the  minute  cal- 
careous crystals  forming  the  skeleton  are  laid  down "  (p.  34). 
Since  he  refers  to  the  skeleton  as  "  ectoplastic  "  in  origin  (p.  113), 
it  is  evident  that  he  agi-ees  with  Bourne  in  the  view  that  the 
organic  matrix  was  not  part  of  the  living  tissues  when  calcareous 
matter  began  to  be  deposited  in  it.  But  in  the  account  of  these 
authors  there  is  no  more  evidence  to  show  that,  in  the  Madre- 
poraria,  the  organic  ground  substance  or  "colloid  matrix  "  (2,  p.  539) 
was  non-living  at  every  phase  of  skeleton  formation  than  that  the 
areas  of  the  scleroblasts  of  the  Alcyonaria  in  which  the  deposition 
of  spicular  matter  took  place  had  not,  at  least  at  the  initial  stages 
of  this  process,  formed  part  of  the  living  protoplasm. 

Further  if,  in  the  Madreporaria,  the  calcareous  matter  were 
deposited  outside  the  living  calicoblastic  ectoderm,  it  is  difficult 
to  understand  how  the  manifold  patterns  of  eoralla  so  charac- 
teristic of  this  gi'oup  of  organisms  can  have  been  built  up*.  But 
if  the  matrix  in  which  carbonate  of  lime  is  laid  down  is  part 
of  the  living  calicoblastic  sheet,  it  follows  that  the  protoplasm 
must  regulate  the  arrangement  of  the  calcareous  matter  into  the 
various  skeletal  types  which,  in  large  measure,  maintain  their  re- 
spective form  independent  of  changes  in  environmental  conditions. 
Similarly,  the  formation  of  the  various  kinds  of  spicules  of  the 
Alcyonaria  can  be  adequately  explained  only  if  calcareous  deposition 
takes  place  within  living  protoplasm,  and  indeed.  Bourne  has  drawn 
attention  to  the  phenomenon  that  "  the  spicules  of  the  Alcyonaria 
show  a  definite  and  complex  crystalline  structure,  the  details  of 
which  are,  indeed,  moulded  upon  and  dominated  by  an  equally 
complex  organic  matrix..."  (2,  p.  517). 

The  intraprotoplasmic  origin  of  spicules  in  the  Alcyonaria  might, 
without  difficulty,  be  ascertained  since  sections  can  be  made  with- 
out decalcification,  whereas  in  Heliopora  and  the  Madreporaria 
possessing  massive  eoralla,  satisfactory  sections  are  possible  only 
after  decalcification,  and  in  this  condition  the  skeleton  may  appear 

*  In  explanation  of  this  phenomenon,  Bourne  suggests  that  "the  general 
arrangement  of  the  fasciculi  of  crystals  is  dominated,  in  some  manner  of  which  we 
are  ignorant,  by  the  living  tissues  which  clothe  the  corallum  "  (2,  p.  539). 

VOL.  XIX.  PART  IV.  12 


162  Mr  Matthai,  Is  the  Madrejjorarian  Skeleton 

as  though  formed  outside  the  living  tissues.  A  further  difficulty 
with  regard  to  the  Madreporaria  is  that,  except  perhaps  at  the 
growing  points,  the  skeleton  would  secondarily  lose  its  intraproto- 
plasmic  character  and  appear  to  be  external  to  the  living  tissues  by 
having  displaced  most  of  the  protoplasm  in  which  it  was  deposited, 
just  as  the  discrete  condition  of  fully  developed  Alcyonarian 
spicules  is  due  to  the  increase  of  calcareous  matter  at  the  expense 
of  the  protoplasm  in  which  it  was  formed. 

From  the  above  considerations  it  would  appear  to  be  highly 
probable  that  von  Heicler  was  right  in  regarding  the  Madreporarian 
skeleton  as  formed  within  the  calicoblastic  protoplasm.  Bourne 
directs  much  of  his  criticism  to  von  Heider's  suggestion  that  the 
striae  in  the  calicoblastic  layer  (i.e.,  in  the  processes  of  attachment) 
are  calcareous  fibres,  but  it  is  not  improbable  that,  in  the  unde- 
calcified  condition,  some  of  these  processes  of  attachment  might 
be  partially  calcified. 

When  thin  sections  of  Astrseid  coralla  are  examined  under  a 
microscope,  they  frequently  appear  to  consist  of  calcareous  pieces 
united  by  sutures  resembling  the  "  laminae  "  or  "  trabecules  "  of  the 
skeleton  of  Heliopora  (1,  p.  463,  pi.  11,  figs.  7  and  8)  and  the  "  tra- 
becular parts  "  of  the  Madreporarian  skeleton  as  figured  by  Ogilvie 
(9,  p.  124,  figs.  13,  19,  etc.).  Each  piece  is  composed  of  calcareous 
strands  radiating  from  a  dark  centre  or  line  which,  as  Ogilvie  sug- 
gested, appears  to  be  the  organic  remains  of  the  protoplasm  in  which 
the  calcareous  needles  were  laid  down.  There  is  some  similarity 
between  these  elements  and  the  spicules  of  Tuhipora  (7,  figs.  9 
and  10)  which,  according  to  Hickson,  are  not  fused  together  but 
dovetailed  into  one  another  as  in  the  membrane  bones  of  Mammals 
(p.  562).  The  resemblance  is  also  marked  in  the  case  of  the  scale- 
like spicules  of  Plumarella  (2,  figs.  6  and  7)  containing  dark  centres 
from  which  calcareous  fibres  or  rods  radiate. 

It  is  difficult  to  gather  from  Bourne's  account  what  he  considers 
to  be  the  unit  of  skeletal  structure  in  the  Alcyonaria.  Are 
spicules  such  units* ?  But  spicules  are  not  all  homologous  elements 
since  they  are  formed  in  protoplasmic  areas  containing  one  or  more 
nuclei  and  no  limit  can  be  set  to  their  size  in  the  various  genera 
(2,  pp.  508-517),  an  extreme  case  being  the  scale-like  spicules  of 
Primnoa  and  Plumarella,  each  of  which  is  "  formed  by  several 
cells,  or  at  least  by  a  comparatively  large  coenocytial  investment 
containing  many  nuclei "  (p.  510).  Or,  is  a  spicule  a  calcareous 
piece  which  behaves  like  a  single  crystal  when  examined  under 
crossed  Nicols?  The  same  confusion  prevails  with  regard  to  ske- 
letal units  in  the  Madreporaria — whether  they  are  represented  by 
"  fibro-crystals  "  (Bourne),  "crystalline  sjjhgeroids"  (von  Koch)  or 

*  Bourne  applies  the  term  spicule  to  "an  entoplastic  product  of  a  single  cell  or 
of  a  ccenocyte  "  (2,  p.  504).   The  italics  are  mine. 


an.  Extra  protoplasmic  Hecretiou  of  the  Polyps  ?  163 

"  calcareous  scales  "  (Ogilvie).  The  latter  are  not  calcified  calico- 
blastic  "cells"  as  Ogilvie  contended  since  the  calicoblastic  ectoderm 
is  now  found  to  be  a  multinucleated  sheet  of  protoplasm  devoid  of 
cell-limits,  i.e.,  a  syncytium. 

In  fact,  there  is  hardly  any  evidence  to  show  that  the  skeleton 
of  the  Anthozoa  is  made  up  of  homologous  units  just  as  it  is  highly 
doubtful  if  their  soft  parts  are  composed  of  uninucleated  units  or 
cells.  The  significance  of  the  Anthozoan  skeleton  would  consist  in 
its  probable  formation  within  syncytial  protoplasm  according  to 
physical  laws  under  the  presiding  activity  of  the  living  protoplasm 
which  would  direct  the  complex  skeletal  architecture.  The  cal- 
careous deposit  further  appears  to  be  differentiated  into  elements 
which  remain  separate  as  spicules  in  most  Alcyonarians  but  are 
united  to  form  a  compact  skeleton  in  certain  Alcyonarians,  e.g., 
Tuhipora,  Corallium,  Heliopora,  and  in  all  the  Madreporaria  (in 
which  the  calcareous  matter  may  undergo  subsequent  rearrange- 
ment). From  this  point  of  view,  a  separate  calcareous  piece  of  an 
Alcyonarian  might  be  regarded  as  a  diminutive  corallum,  and  the 
corallum  of  a  Madreporarian  as  a  massive  spicule,  and  finally,  the 
formation  of  the  Anthozoan  skeleton  would  be  essentially  similar 
to  the  formation  of  membrane  bone  in  Vertebrates*. 

References. 

1.  Bourne,  G.  C.  "  On  the  Structure  and  Affinities  of  Heliopora  ccerulea, 
Pallas.  With  some  observations  on  the  Sti'ucture  of  A'euia  and  Hetero 
xenia."    Phil.  Trans..,  CLXXXVi,  p.  455,  1895. 

2.  Bourne,  G.  C.  "  Studies  on  the  Structure  and  Formation  of  the  Calca- 
reous Skeleton  of  the  Anthozoa."  Quart.  Jour.  Micr.  Sci.,  xli,  p.  499, 1899. 

3.  DuERDEN,  J.  E.  "The  Coral  Siderastrea  radians  and  its  Postlarval 
Development."   Carnegie  Institution,  No.  20,  Washington,  U.S.A.,  1904. 

4.  Fowler,  G.  H.  "  The  Anatomy  of  the  Madreporaria :  I,  Flabelhcm, 
Rhodopsanimia."  Quart.  Jour.  Micr.  Sci.,  xxv,  p.  577,  1885;  and  Stud. 
Owens  Coll.,  I,  p.  243,  1886. 

5.  Heider,  a.  R.  von.  "Die  Gattung  Cladocora,  Ehrb."  Sitzb.  Akad.  Wis- 
sensch.  Wien,  Lxxxiv,  p.  634,  1881. 

6.  Heider,  A.  R.  von.  "  Korallenstudien :  Astroides  calycidaris,  Blainv., 
u.  Dendrophyllia  ramea,  Linn."  Arbeit.  Zool.  Inst.  Graz.  i,  No.  3,  p.  153, 
1886 ;  and  Zeitsch.  Wiss.  Zool.,  XLiv,  p.  507,  1886. 

7.  Hickson,  Sydney  J.  "  The  Structure  and  Relationships  of  Tuhipora.^ 
Quart.  Jour.  Micr.  Sci.,  xxiir,  p.  556,  1883. 

8.  Koch,  G.  von.  "  Ueber  die  Entwicklung  des  Kalkskeletes  von  Asteroides 
Cali/cularis  und  dessen  morphologischer  Bedeutung."  Mitth.  Stat.  Neapel, 
III,  p.  284,  1882. 

9.  Ogilvie,  Maria  M.  "Microscopic  and  Systematic  Study  of  Madreporarian 
Types  of  Corals."   Phil.  Trans.,  clxxxvii,  p.  83,  1896. 

10.    Ogilvie,  Maria  M.     "The  Lime-forming  Layer  of  the  Madreporarian 
Polyp."    Quart.  Jour.  Micr.  Sci.,  XLix,  p.  203,  1906. 

*  It  is  interesting  to  note  that  structures  analogous  to  fibrous  connective  tissue, 
tendon  and  bone  of  Vertebrates,  occur  in  the  Madreporaria,  viz.,  the  middle  lamina 
(  =  mesoglfea),  processes  of  attachment  and  the  calcareous  coraUum,  a  matter  which 
will  be  discussed  in  a  future  communication. 

12—2 


164  Mr  Matthai,  On  Reactions 


On  Reactions  to  Stimuli  in  Corals.  By  G.  Matthai,  M.A., 
Emmanuel  College,  Cambridge.  (Communicated  by  Professor 
Stanley  Gardiner.) 

[Read  18  February  1918.] 

The  following  is  a  brief  record  of  feeding-experiments  made  on 
living  Astrseid  colonies  during  a  short  stay  at  the  Carnegie  Bio- 
logical Station  at  Tortugas  (July  16 — Aug.  2)  and  at  the  Bermuda 
Biological  Station  on  Agar's  Island  (Aug.  20 — Sep.  14)  in  the 
summer  of  1915,  which,  though  necessarily  incomplete  as  they  had 
to  be  undertaken  in  the  midst  of  other  work,  gave  some  indication 
of  the  nature  of  reactions  to  stimuli  in  the  Madreporaria.  In  order 
to  watch  the  behaviour  of  living  Corals,  colonies  of  most  of  the 
recent  species  recorded  from  those  localities  were  kept  in  aquaria 
of  running  sea- water,  viz. : 

Mceandra  lahyrinthifo7^mis  (Linn.),  Moeandra  strigosa  (Dana), 
McBandra  clivosa  (Ell.  and  Sol),  Manicina  areolata  (Linn.),  Colpo- 
phyllia  gyrosa  (Ell.  and  Sol),  Isophyllia  dipsacea  (Dana),  Isophyllia 
fragilis  (Dana),  Dichocoenia  Stokesi,  Ed.  and  H.,  Easrnilia,  aspera 
(Dana),  Favia  fragum  (Esp.),  Orhicella  cavernosa  (Linn.),  Orbicella 
annidaris  (Ell.  and  Sol.),  Stephanocoenia  intersepta  (Esp.),  Ocidina 
diffusa,  Lam.,  Mycetophyllia  lamarckana,  Ed.  and  H.,  Siderastrcea 
radians  (Pallas),  Siderastrcea  siderea  (Ell.  and  Sol.),  Agaricia 
purpurea,  Les.,  Porites  astreoides,  Lam.,  Porites  furcata,  Lam., 
Porites  clavaria,  Lam.,  Madracis  decactis  (Ly.),  and  Acropova 
muricata  (Linn.). 

In  Isophyllia  dipsacea  (Dana),  when  a  particle  of  meat  was 
placed  on  the  oral  disc  with  contracted  mouths,  the  oral  lip 
was  slowly  directed  towards  the  particle  and  the  mouth  became 
dilated,  to  an  extent  depending  on  the  size  of  the  food-particle. 
The  latter  was,  in  the  meantime,  slowly  moved  into  the  oral  open- 
ing by  ciliary  action.  To  facilitate  this  event,  the  periphery  of  the 
oral  disc  was  drawn  over  towards  the  dilated  mouth  and  the  disc 
itself  was  somewhat  depressed,  thus  deepening  the  peristomial 
cavity.  During  distention  of  the  mouth,  the  stomodgeum  was  everted 
and,  consequently,  the  coelenteric  cavity  Avith  its  convolutions  of 
mesenteries  became  exposed.^  After  the  food-particle  had  passed 
into  the  coelenteric  cavity,  it  was  caught  in  the  mesenterial  coils. 
If  the  fragment  of  meat  was  large,  the  mouth  remained  widely  open 
till  the  former  had  been  reduced  in  size  by  the  digestive  action  of 
the  mesenterial  filaments.  The  stomodtEum  was  subsequently  with- 
drawn and  the  mouth  opening  gradually  narrowed.  But  if,  before 
this,  the  oral  lip  was  touched  with  a  glass  needle,  it  did  not  contract 
as  it  would  do  instantaneously  if  no  food-particle  had  previously 


to  Stimuli  in  Corals  165 

been  swallowed.  Every  mouth  that  was  tested  could  thus  take  in 
particles  of  meat.  The  touch  of  the  food-particle  on  the  oral  disc 
was  also  a  stimulus  for  the  expansion  of  the  tentacles  around  the 
mouth  and  of  those  around  the  neighbouring  oral  openings. 

When  a  particle  of  meat  was  placed  on  the  tentacles  of  a  colony 
of  Mceandra  labyrinthiformis  (Linn.),  it  was  slowly  passed  on  to  the 
oral  disc,  but  the  tentacles  did  not  show  any  sign  of  contraction. 
At  the  same  time,  the  oral  disc  was  depressed  and  arched  over  the 
mouth  opening  till  finally  its  margin  closed  over  the  peristome.  In 
the  meantime,  the  tentacles  were  fully  distended,  the  entocoelic 
ones  were  directed  obliquely  towards  the  oral  opening,  those  of 
one  side  passing  between  those  of  the  opposite  side.  The  food- 
particle  was  now  hidden  from  view.  After  it  had  passed  into  the 
ccjelenteric  cavity  and  had  presumably  undergone  partial  digestion, 
the  periphery  of  the  oral  disc  gradually  moved  outwards  carrying 
the  tentacles  with  it,  thus  again  exposing  the  peristomial  cavity. 

The  principal  movements  in  these  two  cases  are: 

(1)  Ciliary  movement  passing  the  food-particle  into  the  nearest 
oral  aperture. 

(2)  The  direction  of  the  oral  lip  towards  the  food-particle  pari 
passu  with  the  dilatation  of  the  mouth. 

(3)  The  narrowing  and  deepening  of  the  peristomial  cavity, 
which  help  to  roll  the  food-particle  into  the  oral  opening. 

(4)  The  expansion  of  the  tentacles  of  the  affected  oral  disc  and 
of  those  of  adjacent  oral  discs. 

(5)  The  eversion  of  the  stomodeeum  and  consequent  exposure 
of  the  coelenteric  cavity  and  mesenterial  coils. 

(6)  The  return  of  the  soft  parts  to  their  original  condition  by 
the  retraction  of  the  stomodseum  into  the  coelenteric  cavity,  recoil 
of  the  oral  lip  to  its  normal  extent,  shortening  of  the  tentacles, 
flattening  of  the  oral  disc  and  withdrawal  of  its  periphery  carrying 
the  tentacles  outwards. 

When  a  drop  of  meat-juice  was  gently  placed  on  a  colony  of 
Favia  frag  am  (Esp.),  the  oral  apertures  in  the  neighbourhood  were 
slowly  distended  after  a  short  pause.  The  inner  or  entocoelic  row  of 
tentacles  was  then  extended  and  directed  over  the  oral  disc,  meeting 
or  intercrossing  over  the  mouth  as  had  been  noticed  in  the  case  of 
Mceandra  labi/rinthiformis  (Linn.),  thus  hiding  the  oral  region, 
whilst  the  exocoelic  tentacles  were  arched  outwards.  Similar  move- 
ments were  observed  in  Mceandra  strigosa  (Dana). 

When  meat-juice  was  spurted  by  a  pipette  on  sea- water  con- 
taining a  colony  of  Orhicella  cavernosa  (Linn.),  strong  contraction 
of  the  soft  parts  was  set  up  in  the  neighbourhood,  the  polyps  en- 
tirely closing  up.  This  was  followed  by  the  protrusion  of  convolutions 
of  mesenteries  through  mouth  openings,  oral  discs  and  especially 
through  edge-zones,  combined  with  secretion  of  mucus  over  the 
polyps,  the  former  obviously  to  paralyse  prey  and  the  latter  to 


166  Mr  Matthai,  On  Reactions  to  Stimuli  in  Corals 

entangle  food-particles.  Shortly  afterwards,  the  oral  apertures  were 
widely  distended  to  let  in  the  meat-juice  but  the  process  was  un- 
accompanied by  eversion  of  stomodsea.  Similar  events  were  observed 
in  Manicina  aj-eolata  (Linn.). 

When  finely  powdered  carmine  was  scattered  in  sea-water  con- 
taining a  colony  of  Manicina  areolata  (Linn.),  it  was  partly  taken 
into  the  stomoda^a,  the  oral  lips  becoming  conspicuously  stained. 
The  carmine  was,  however,  subsequently  passed  out  of  the  stomodaea, 
showing  thereby,  that  the  mouth  openings  could  function  as  in- 
halent  and  exhalent  apertures. 

When  a  tentacle  of  any  of  the  Astraiid  colonies  was  touched 
with  a  fine  glass  needle,  it  was  suddenly  withdrawn  in  a  manner 
resembling  pseudopodial  movement  and  the  neighbouring  tentacles 
were  also  retracted.  In  Porites  and  Madracis,  whose  soft  parts  are 
composed  of  small  polyps,  the  instantaneous  contraction  of  a  polyp 
due  to  mechanical  stimulation  caused  the  contraction  of  its  neigh- 
bours as  well.  In  all  these  cases,  the  wave  of  contraction  started 
from  a  centre,  viz.,  the  point  of  stimulation,  but  remained  local  and 
did  not  spread  over  the  entire  colony. 

Series  of  movements  such  as  the  above,  made  in  response  to 
chemical  and  tactile  stimuli,  are  reminiscent  of  amoeboid  or  stream- 
ing movement  of  protoplasm,  the  soft  parts  of  the  colonies  appearing 
to  serve  as  the  medium  for  the  transmission  of  stimuli*.  If  the 
initial  stimulus  be  too  strong,  the  sudden  contraction  of  the  soft 
parts,  due  to  the  mechanical  impact,  is  followed  by  slow  purposive 
movements. 

The  amoeboid  character  of  the  movements  of  the  soft  parts  of 
Astrseid  Corals  is  in  conformity  with  their  histological  structure 
which,  on  examination,  revealed  neither  a  muscular  nor  a  nervous 
system,  although  a  neuro-muscular  apparatus  has  been  supposed 
by  most  authors  to  exist  in  Madreporaria.  The  so-called  muscular 
fibres  at  the  base  of  the  ectoderm  and  endoderm  seem  to  be  of  the 
nature  of  specialised  connective  tissue  fibres,  for  in  both  teased 
preparations  and  in  sections  of  4/z — 10/i  thicknesses  these  are  found 
to  be  without  nuclei  and  to  form  part  of  the  middle  lamina  (=  meso- 
glsea)  which  is  itself  composed  of  fine  fibres  cemented  together  by 
a  homogeneous  matrix  containing  a  few  scattered  nucleated  cells. 
Fibrils  pass  into  the  middle  lamina  through  the  granular  stratum 
present  at  the  base  of  the  ectoderm  (and  less  frequently  at  the  base 
of  the  endoderm),  but  these  fibrils  do  not  show  any  histological 
differentiation  which  would  justify  us  in  regarding  them  as  belong- 
ing to  nerve  elements f. 

*  Carpenter  I'egarded  the  feeding  reactions  of  Isophyllia  as  muscular  in  nature 
and  as  brought  about  by  the  transmission  of  impulses  of  a  "  nervoid  character," 
but  he  had  not  investigated  the  histological  structure  of  its  soft  parts  {vide  Con- 
tributions Bermuda  Biol.  Station,  No.  20,  Cambridge,  Mass.,  U.S.A.,  p.  149,  1910). 

t  For  a  detailed  account  of  the  minute  structure  of  coral  polyps  vide  "The 
Histology  of  tlie  Soft  Parts  of  Astraeid  Corals  "  to  be  published  shorth'. 


Mr  Brindley,  Notes  on  certain  parasites,  food,  etc.         167 


Notes  on  certain  parasites,  food,  and  capture  hy  birds  of  the 
Common  Earwig  (Foi-ficula  aiiricularia).    By  H.  H.  Brindley,  M.A., 

St  John's  College. 

[Read  18  February  1918.] 

(rt)  Effects  of  pa7'asitism. 

In  a  paper  entitled  "  The  effects  of  Parasitic  and  other  kinds 
of  castration  in  Insects "  (Jour.  Exper.  Zool.  viii.  Philadelphia, 
1910)  Wheeler  expresses  the  opinion  (p.  419)  that  Giard  has  given 
good  reasons  for  supposing  that  the  dimorphism  exhibited  by  the 
forcipes  of  male  earwigs  from  the  Farn  Islands,  Northumberland 
(Bateson  and  Brindley,  "  On  some  cases  of  variation  in  secondary 
sexual  characters  statistically  examined,"  Proc.  Zool.  Soc.  Lond. 
1892,  p.  585),  is  due  to  "differences  in  the  number  of  gregarines 
they  harbour  in  their  alimentary  tract."  The  reference  to  Giard 
is  C.R.  Acad.  Sci.  cxviii.  1894,  p.  872,  where  he  writes  "  J'ai  tout 
lieu  de  croire  qu'une  interpretation  du  meme  genre  (referring  to 
the  changes  evoked  in  Carcinus  by  the  action  of  parasites)  pent 
s'appliquer  pour  la  distribution  des  longueurs  des  pinces  des 
Foi'ficules  males.  II  est  possible,  en  effet,  d'apres  la  longueur  de 
la  pince,  de  prevoir  qu'une  Forficule  male  possede  des  Gregarines 
et  qu'elle  en  possede  une  plus  ou  moins  grande  quantite." 

In  criticism  of  the  above  statements  Capt.  F.  A.  Potts  and 
myself  published  a  letter  in  Science,  Philadelphia,  Dec.  9,  1910, 
p.  836,  in  which  we  gave  reasons  for  disagreeing  with  Wheeler's 
conclusion  :  viz.,  (i)  that  in  the  absence  of  any  further  account  by 
Giard  the  above  passage  could  not  be  taken  as  direct  evidence 
that  he  had  examined  the  intestine  of  Forficula  for  gregarines  and 
found  a  correspondence  between  their  presence  and  the  condition 
of  the  male  forcipes ;  (ii)  that  out  of  several  thousand  earwigs 
collected  by  us  on  the  Farn  Islands  in  1907  over  50  males  of 
different  forceps  lengths  were  carefully  dissected  with  the  results 
that  the  gregarine  Clepsydrina  ovata  was  found  to  occur  commonly 
in  the  alimentary  canal,  that  it  occurred  indifferently  and  was 
absent  indifferently  in  "  low "  and  "  high "  males,  and  that 
no  correlation  could  be  traced  between  the  number  of  parasites 
and  the  length  of  its  forcipes.  Moreover,  no  difference  in  the 
development  of  the  testes  or  other  internal  sexual  organs  could 
be  detected  in  low  and  high  males  respectively. 

Since  the  above  was  written  I  have  (August  1917)  examined 
the  alimentary  canal  of  51  earwigs  out  of  a  large  batch  obtained 
at  Porthcressa,  St  Mary's,  Isles  of  Scilly,  where  the  males  exhibit 


168     Mr  Brindley,  Notes  on  certain  parasites,  food,  and  capture 

well-marked  dimorphism  (Camb.  Phil.  Soc.  Proc.  xvii.  part  4, 1914, 
p.  831). 

The  results  summarised  are  as  follows: 


Infection  by  Clepsydrina  ovata. 


Number 
examined 

Not 
infected 

Infected 

Number  of 

gregarines 

found 

Average  number 

of  gregarines  in 

the  infected 

individuals 

Low  males 
High  males 
Females 

23 
23 

5 

12 

11 

1 

11 
12 

4 

323 

238 
53 

29 
20 
13 

Thus  the  evidence  so  far  obtained  is  that  the  dimorphism  of 
the  forcipes  in  F.  auricularia  </  is  not  a  result  of  or  influenced  by 
gregarine  infection — though  in  view  of  the  well-established  effects 
of  such  parasitism  on  the  secondary  sexual  characters  of  another 
arthropod  in  Geoffrey  Smith's  case  o^ Inachus  dorsettensis  modified 
by  the  gregarine  Aggregata  {Mitt.  Zool.  Stat.  Neap.  xvii.  1905, 
p.  406),  the  absence  of  positive  evidence  to  the  contrary  at  the 
time  Wheeler  wrote,  but  now  obtained,  certainly  afforded  ground 
for  his  support  of  Giard. 

In  this  connection  I  may  quote  a  letter  from  Geoffrey  Smith, 
whose  recent  death  at  the  battle  front  brings  us  into  common 
mourning  with  Oxford  zoologists  for  a  friend  and  colleague. 
Writing  to  me  about  1907  he  said,  "  Have  you  noticed  that  Giard 
attributes  all  cases  of  High  and  Low  Dimorphism  to  parasitic 
castration  ?  I  am  sure  this  is  not  right,  but  there  is  no  doubt 
that  parasitic  castration  is  a  much  more  frequent  occurrence  than 
is  commonly  supposed."  These  words,  and  a  footnote  to  the  same 
effect  in  his  paper  "  High  and  Low  Dimorphism  "  (Mitt.  Zool.  Stat. 
Neap.  XVII.  1005,  p.  321),  are  typical  of  the  writer's  insight  and 
balanced  judgment. 

It  may  be  stated  that  the  gregarines  in  the  Porthcressa  earwigs 
fell  roughly  into  categories  of  small,  medium,  and  large,  but  they 
all  seemed  to  be  C.  ovata.  Rather  more  than  half  were  small 
individuals,  and  those  of  medium  size  were  slightly  in  excess  of  the 
large,  but  the  sizes  were  not  recorded  in  the  case  of  the  first  few 
earwigs  examined.  Very  large  numbers  were  found  in  syzygy, 
and  such  associated  individuals  were  of  all  three  sizes.  One 
instance  of  syzygy  of  a  large  with  quite  a  small  individual  was 
observed.      There   was    no    noteworthy   difference    between   the 


hy  birds  of  the  Common  Earwig  (Forficula  auriculana)     169 

numbers  of  gregarines  of  different  sizes  or  between  the  proportion 
of  free  gregarines  to  those  in  syzygy  in  their  low  and  high  male 
hosts  respectively. 

During  our  stay  on  the  Scilly  Islands  in  1912  Capt.  Potts  and 
myself,  in  company  with  Capt.  J.  T.  Saunders,  found  in  St  Martin's 
several  earwigs  parasitised  by  a  gordiid  larva  {sp.  incert.),  the  coils 
of  which,  though  projecting  between  the  terga  of  the  abdomen, 
seemed  to  have  no  effect  on  the  health  and  activity  of  their  hosts. 
The  same  apparent  absence  of  deleterious  effects  was  noticed  in 
three  of  the  Porthcressa  batch  of  1917  which  were  found  to  be 
similarly  infected.  In  one,  a  low  male,  a  large  gordiid  occupied 
most  of  the  body,  and  no  portion  of  the  alimentary  canal  posterior 
to  the  crop  could  be  found ;  in  a  high  male  similarly  infested  by  a 
large  gordiid  there  was  very  little  of  the  hind  gut  left ;  and  an  adult 
female  contained  three  or  four  gordiids  of  various  sizes,  the  gut  in 
this  case  being  intact  and  apparently  healthy.  A  fourth  individual, 
a  low  male,  was  not  parasitised  when  examined,  but  as  the  gut  was 
partially  atrophied,  it  had  probably  been  recently  deserted  by  a 
gordiid.  All  these  infected  individuals  seemed  as  active  and 
healthy  and  to  possess  fat  bodies  as  large  as  those  not  infected  ; 
the  earwig's  resistance  to  such  extensive  destruction  of  internal 
organs  is  very  noteworthy.  As  Clepsydrina  ovata  inhabits  the 
chylific  ventricle  and  hind  gut  and  as  the  presence  of  gordiids 
evidently  often  results  in  destruction  of  these  portions  of  the 
alimentary  tract,  the  latter  parasite  is  likely  to  be  exclusive  of 
gregarines,  and  these  were  absent  in  all  three  of  the  males 
mentioned  above  (including  that  with  the  hind  gut  intact),  while 
only  two  were  found  in  the  female. 

That  the  presence  of  parasitic  worms  has  sometimes  serious 
effects  on  the  insect's  health  is  suggested  by  the  recent  observations 
of  Jones  recorded  in  "  The  European  Earwig  and  its  control," 
a  report  on  the  invasion  of  Newport,  R.I.,  in  1911  by  Forficida 
auricularia  and  its  subsequent  spread  ( f/. >§.  Dept.  Agric.  Bidl.  566, 
Washington,  June,  1917),  from  which  it  appears  that  10  per  cent, 
of  earwigs  kept  in  the  laboratory  were  killed  by  the  infection  of  a 
worm  identified  as  Filaria  locustae,  whose  average  length  is  given 
as  83  mm.  This  however  is  a  size  exceeding  considerably  that  of 
the  gordiids  in  the  Scilly  earwigs,  which  I  have  called  "  large " 
when  attaining  a  length  of  50  mm. 

In  southern  Russia  Forficula  tomis,  Kolenati,  is  parasitised  by 
the  tachinid  fly,  Rhacodineura  antiqua  (Pantel,  Bull.  Soc.  Entom. 
France,  No.  8,  Paris,  1916,  p.  150),  but  I  do  not  know  if  it  attacks 
the  common  earwig.  The  paper  quoted  mentions  the  capture  of 
the  adult  fly  in  Holland  and  Portugal. 

Lucas  {Entom.  XXXVII.  1904,  p.  213)  reports  F.  auricularia 
(or  ?  lesnei)  attacked  by  scarlet  acarine  mites. 


170     Mr  Brindley,  Notes  on  certain  parasites,  food,  and  capture 

Among  fungoid  parasites,  EntomopMhora  forficulae  diminishes 
the  number  of  earwigs  (Picard,  Bidl.  Soc.  Etude  Vulg.  Zool.  Agric. 
Bordeaux,  Jan. — April,  1914,  pp.  1,  25,  37,  62).  It  is  possibly  this 
species  which  has  caused  heavy  mortality  among  the  earwigs  which 
I  have  kept  in  captivity  in  the  Zoological  Laboratory  during 
recent  years.  Infection  by  the  above  or  other  fungus  is  a  very 
frequent  result  of  damp  in  the  soil  or  in  the  plaster  of  Paris  cells 
bedded  with  coco  fibre  which  I  have  employed.  The  most  effective 
preventive  of  fungus  has  so  far  been  keeping  the  earwigs  in 
roomy  glass  dishes  lined  with  virtuall}^  dry  sand  and  supplj^ing 
water  only  by  wetting  the  vegetable  food  given. 

(6)   Food. 

In  "  The  Wild  Fauna  and  Flora  of  the  Royal  Botanic  Gardens, 
Kew,"  1906  {Kew  Bull.  Add.  Series  V),  Lucas  writes  (p.  23)  of  the 
Common  Earwig,  "  It  is  an  animal  feeder.  Does  it  do.  as  much 
damage  as  is  supposed  ? "  And  Ealand  in  "  Insects  and  Man," 
1915,  p.  266,  states  "most  gardeners  would  assert  that  the  insect 
is  destructive  to  cultivated  plants.  Careful  observation  and 
experiment,  however,  show  that  it  is  carnivorous  and  that  it 
devours  caterpillars,  snails,  slugs,  etc.... its  habit  of  hiding  in  such 
flowers  as  the  sunflower  and  dahlia  have  earned  it  an  undeserved 
reputation  for  evil." 

I  find  that  seven  out  of  nine  recent  and  more  or  less  compre- 
hensive manuals  of  Economic  Entomology  do  not  mention  earwigs 
at  all,  which  is  fair  evidence  for  considerable  doubt  as  to  their 
being  harmful  insects.  Of  the  two  works  in  which  earwigs  are 
mentioned  one  speaks  of  them  as  destructive  to  mangolds,  turnips, 
cabbage  crops,  and  plant  blossoms,  while  the  other  states  dahlias 
as  attacked,  "  but  nearly  all  plants  suffer."  Virtually  every  fruit 
grower  and  horticulturist  of  whom  we  make  enquiry  assures  us 
that  earwigs  are  most  destructive  pests,  but  is  the  general  belief 
thus  expressed  really  well  founded  ? 

Recent  literature  leaves  the  impression  that  in  certain  localities 
earwigs  may  be  specially  harmful  to  plants  of  economic  value, 
though  an  explanation  of  this  capriciousness  is  wanting.  Theobald 
(Rep.  on  Econ.  Zool.,  South-Eastern  Agric.  Coll.,  Wye,  April  1914) 
gives  hops  as  attacked  by  F.  auricidaria.  Lind  and  others  in  a 
summary  of  the  diseases  of  agricultural  plants  in  1918  (79  Be- 
retning  fra  Staiens  Forsogsvirksamded  i  Plantekidtur,  no.  30, 
Copenhagen,  1914)  state  that  in  one  locality  in  Denmark  cauli- 
flowers were  completely  destroyed  by  the  Common  Earwig,  which 
seems  a  very  exceptional  event.  Sch^^iyen  in  Beretning  om  skadein- 
sekter  og plantesygdommer  i  land  og  havchruket  1915  (Report  on  the 
injurious  insects  and  fungi  of  the  field  and  the  orchard  in  1916), 


hy  birds  of  the  Common  Earivig  (Forficula  auricnlaria)     171 

Kristiania,  1916,  mentions  that  in  many  parts  of  Norway  different 
vegetables,  cabbage  in  particular,  were  extensively  damaged  by 
F.  auricularia.  Tullgren,  in  a  report  on  injurious  animals  in 
Sweden  during  1912 — 1916  (Aleddelande  frdn  Centrcdanstalten 
for  Jorshruksforsok,  no.  152 ;  Entomologiska  Avdelningen,  no.  27, 
p.  104),  records  damage  by  F.  auricidaria  to  ornamental  plants, 
barley,  wheat,  and  cabbage.  In  the  case  of  the  invasion  of  New- 
port, R.I.,  by  the  Common  Earwig,  Jones  {op.  cit.)  reports  that  the 
quite  young  individuals  eat  tender  shoots  of  clover  and  grass,  and 
possibly  grass  roots  ;  while  later  on  shoots  of  Lima  Bean  and  dahlia 
and  blossoms  of  Sweet  William  and  early  roses  are  attacked,  with 
a  general  preference  for  the  bases  of  petals  and  stamens  rather 
than  for  green  shoots.  Adults  are  recorded  as  feeding  almost  wholly 
on  petals  and  stamens,  though  clover,  grass  and  terminal  buds  of 
chrysanthemums  and  other  "fall  flowers"  are  also  devoured.  Sopp, 
"The  Callipers  of  Earwigs"  {Lanes,  and  dies.  Entom.  Soc.  Proc. 
1904,  p.  42),  records  having  seen  a  female  earwig  using  her  forcipes 
to  repeatedly  pierce  damp  decaying  seaweed  on  which  she  was 
apparently  feeding.  Ltistner  {Centralhl.  Bakt.  Parnsit.  u.  Infektions- 
krankheiten,  XL.  nos.  19-21,  Jena,  April  1914,  p.  482)  has  summa- 
rised the  work  of  over  thirty  observers  of  the  contents  of  the  crop 
of  the  Common  Earwig.  Altogether  162  individuals  were  thus 
examined,  and  the  conclusion  was  arrived  at  that  earwigs  normally 
feed  on  dead  portions  of  plants  and  on  fungi  such  as  Gapnodium, 
living  leaves  and  flowers  being  attacked  when  circumstances 
favoured  the  change.  Dahlia  leaves  and  petals  were  very  readily 
devoured.  How  far  earwigs  are  a  pest  to  ripe  fruit  seems  not  to 
have  been  investigated,  but  it  was  concluded  that  as  a  rule  they 
may  be  regarded  as  harmless  save  in  special  cases.  It  was  admitted 
however  that  the  further  the  enquiry  went  the  less  definite  were 
the  results. 

In  view  of  the  diversity  of  reports  as  to  the  favourite  food 
plants  of  earwigs  and  the  general  want  of  exact  information  as  to 
the  damage  likely  to  be  done  by  earwigs  in  a  flower  or  kitchen 
garden  I  carried  out  a  small  series  of  observations  on  the  earwigs 
obtained  last  August  from  St  Mary's,  Isles  of  Scilly,  which  were 
kept  in  captivity  in  the  Zoological  Laboratory  for  some  weeks, 
primarily  for  the  purpose  oj"  examining  their  alimentary  canal  for 
parasites.  These  earwigs,  several  dozen  in  number,  were  kept  in 
a  large  glass  dish  bedded  with  sand  slightly  damped  occasionally. 
They  had  no  animal  food  save  that  afforded  by  those  which  died. 
In  order  to  obtain  information  as  to  preference  for  one  kind  of 
plant  above  another  they  were  given  three  different  species,  taken 
haphazard,  at  a  time  for  a  period  of  two  days  or  more. 

A  summary  of  the  results  is  as  follows :  — 

Aug.  20  and  21.     Vegetable  marrow  leaves  were  ver\'  much 


172     ilf?'  Brindley,  Notes  on  certain  parasites,  food,  and  capture 

eaten ;  horse-radish  leaves  very  little  touched ;  Michaelmas  Daisy 
leaves  and  flowers  hardly,  if  at  all,  touched. 

Aug.  22  and  23.  Beetroot  leaves  were  much  eaten,  the  leaf 
stalks  m  particular,  these  being  opened  out  and  the  pith  taken : 
white  phlox  leaves  and  flowers,  the  petals  much  gnawed  and  pollen 
grains  were  found  in  the  gut :  dwarf  bean  leaves,  little  touched. 

Aug.  24  to  26.  Blue  Anchusa  leaves  and  flowers,  the  petals 
were  much  eaten  but  the  leaves  neglected :  white  rose  leaves  and 
flowers,  petals  devoured  but  leaves  untouched:  golden  rod  (Solidago) 
leaves  and  flowers,  leaves  nibbled  at  sides  here  and  there  but 
flowers  apparently  neglected. 

Aug.  27  to  29.  Yellow  Oenothera  flowers  and  pods,  the  petals 
were  much  eaten  but  the  pods  remained  untouched  :  white  Japanese 
anemone  leaves  and  flowers,  petals  eaten  to  some  extent,  leaves 
neglected :  raspberry  foliage,  the  leaves  were  not  nibbled,  but  the 
earwigs  congregated  in  numbers  on  their  hairy  undersides,  an 
action  much  more  pronounced  than  in  the  case  of  any  of  the  other 
plants  given  throughout  the  observations. 

Aug.  30  and  31.  Cabbage  leaves  were  destroyed  by  the  blade 
bemg  gnawed  down  between  the  veins  to  the  midrib  while  the 
ends  of  the  veins  were  shorn  off:  rhubarb  leaves,  eaten  a  good 
dea  :   scarlet  runner  leaves,  flowers,  and   pods,  apparently  quite 

Sept.  1  to  3.  Plum  fruit  unskinned  was  much  attacked- 
potato  tuber  and  rather  unripe  apple,  both  unskinned,  were  not 
touched  at  all. 

Sept  4  to  10.  On  the  4th  the  plum  was  removed,  but  the 
apple  and  potato  were  not  attacked  during  the  seven  days. 

Sept  11  to  15.  On  the  11th  the  apple  was  cut  across,  with 
the  result  that  it  was  slightly  gnawed  during  the  five  days :  the 
potato  remained  untouched. 

Sept.  16  to  2:l  On  the  16th  the  potato  was  cut  across,  which 
was  followed  by  its  being  very  thoroughly  attacked,  though  the 
apple  was  not  entirely  deserted. 

Of  the  51  earwigs  whose  alimentary  canals  were  examined  for 
gregarine  7  contained  spores  of  Fuccinea  graminis  (one  had  as 
many  as  180  and  another  100),  while  the  food  of  another  individual 
included  numerous  unidentified  enjiomophilous  pollen  grains 
Both  spores  and  pollen  grains  appeared  to  be  very  slightly  if  at  all" 
digested.  It  is  hoped  to  extend  the  observations  in  the  coming 
summer,  as  those  recorded  above  were  limited  to  only  a  few  of  the 
possible  food  plants  and  only  adult  earwigs  were  kept.  It  may 
well  be  that  there  are  differences  in  the  preferences  of  nymphs 
and  adults,  and  as  the  former  are  in  the  majority  till  about  the 
end  of  July,  it  is  possible  that  they  may  be  harmful  to  certain 
plants  m  particular,  as  Jones's  observations  (o;j>.  cit.)  suggest. 


hy  birds  of  tJie  Covinion  Eariuig  (Forficula  auricularia)     173 

It  seems  established  that  a  large  number  of  ordinary  garden 
species  are  liable  to  serious  attack  by  earwigs,  and  that  the  latter 
can  continue  healthy  on  a  purely  vegetable  diet.  But  much  further 
information  of  a  detailed  kind  is  required  befoi'e  we  can  explain 
why  in  a  given  locality  a  particular  kind  of  plant  is  attacked 
while  in  another  it  is  neglected.  Does  it  mean  that  the  presence 
or  absence  of  suitable  animal  food  is  a  factor  ? 

As  regards  animal  food,  there  is  a  considerable  amount  of 
evidence  that  earwigs  are  often  carnivorous  by  choice,  very 
possibly  they  are  so  usually  (cf  Rlihl,  M.T.  Schweiz.  Ges.  vii. 
1887,  p.  310).  In  respect  of  eating  dead  animal  matter  I  have 
found  that  when  kept  in  captivity  they  devour  the  soft  parts 
of  their  fellows  who  have  died  even  when  fresh  vegetable  food 
is  available.  In  this  necrophagous  habit  they  resemble  cock- 
roaches. Jones  {op.  cit.)  states  that  dead  flies  and  dead  or  dying 
comrades  are  devoured.  Lustner  (op.  cit.)  finds  that  only  dead 
animal  matter  is  taken.  This  conclusion  points  to  too  limited  an 
inquiry  and  want  of  taking  into  account  the  possible  presence  of 
food  plants  which  were  more  attractive  than  available  living  prey. 
In  any  case  his  opinion  that  earwigs  should  not  be  regarded  as 
beneficial  is  traversed  by  the  records  of  their  killing  certain  insect 
pests  of  plants. 

Round  Island,  the  northernmost  islet  of  the  Scilly  group,  is 
swarming  with  earwigs,  and  they  congregate  in  vast  numbers  in 
the  light-keepers'  midden  inside  the  discarded  pressed  beef  tins. 
If,  as  seems  probable,  they  reached  the  islet  before  the  lighthouse 
was  built  a  change  of  diet  seems  to  have  occurred,  as  the  indigenous 
vegetation  is  chiefly  Armeria  maritima,  Cochlearia  officinalis  and 
Mesembryanthemum  edide.  There  is  no  turf  It  is  of  course 
possible  that  they  seek  the  potato  peelings  also  thrown  into  the 
midden  and  that  their  numbers  inside  the  discarded  tins  mean 
that  the  latter  are  frequented  partly  for  shelter.  If  the  Round 
Island  earwigs  have  really  turned  during  comparatively  recent 
years  from  a  herbivorous  to  an  extensively  carnivorous  diet, 
Rosevear,  another  islet  of  the  Scilly  group  may,  in  a  sense,  be  a 
converse  case.  It  is  the  other  locality  in  the  Scilly  group  in  which 
(as  far  as  I  know)  the  earwig  population  is  densest.  Like  Round 
Island,  it  is  very  small,  but  differs  from  it  in  being  uninhabited. 
But  from  1850  to  1858  it  was  occupied  by  the  builders  of  the 
Bishop  Rock  Lighthouse,  so  is  it  possible  that  the  abundance  of 
earwigs  is  due  to  the  animal  food  available  in  the  past  ?  However 
this  may  be  the  present  diet  of  the  Rosevear  earwigs  appears  likely 
to  be  vegetarian  in  the  main,  unless  the  islet  harbours  some  insect 
or  other  small  arthropod  suitable  for  food.  The  commonest  plants 
are  Armeria  maritima  and  Lavatera  arborea,  the  latter  growing 
luxuriously.     But  before  the  abundance  of  earwigs  on  Rosevear 


17-i     Mr  Brindley,  Notes  on  certain  jKtrasites,  food,  and  cajytare 

can  be  discussed  adequately  something  must  be  known  of  the  con- 
ditions obtaining  on  Rosevean  and  Gorregan,  its  small  and  only 
immediate  neighbours.  Of  these  islets  I  possess  no  information 
at  present.  Also,  there  are  other  peculiarities  as  regards  the 
earwigs  of  Rosevear  and  Round  Island  which  are  beyond  the 
scope  of  the  present  paper. 

There  is  no  doubt  that  earwigs  sometimes  kill  and  devour 
other  insects  larger  than  themselves,  though  the  event  is  probably 
somewhat  exceptional.  Chapman  ("Notes  on  Early  Stages  and 
Life  History  of  the  Earwig,"  Entom.  Record,  xxix.  no.  2,  Jan. 
1917)  states  that  "animal  food,  such  as  dead  insects,  seemed  always 
acceptable  "  to  earwigs  in  captivity.  Sopp  {op.  cit.  p.  42)  regards 
earwigs  as  probably  "omnivorous  feeders,  largely  carnivorous  by 
choice,  but  often  phytophagous,  frugivorous,  or  even  necrophagous 
of  necessity."  Whether  attack  on  living  animals  as  prey  is 
common  I  cannot  say,  I  have  no  observations  of  my  own  to 
record ;  ^  but  it  appears  that  occasionally  the  forcipes,  organs  of 
much  disputed  function,  are  used  for  this  purpose.  Sopp  (op.  cit.) 
has  seen  them  employed  to  seize  and  crush  large  flies  which  were 
.  subsequently  devoured  and  quotes  an  instance  of  a  larva  similarly 
attacked  from  the  records  of  another  observer.  Burr  (Entom. 
Record,  Sept.  1903)  saw  a  blue-bottle  seized  by  the  forcipes  of  a 
male  Labidura  riparia  kept  in  captivity.  Lucas  {Entom.  xxxviii. 
1905,  p.  267)  records  a  female  of  this  species  as  using  the  forcipes 
to  capture  a  cinnabar  moth  larva,  which  was  afterwards  devoured. 
Jones  {op.  cit.)  records  that  the  Newport,  R.I.,  earwigs  attack  and 
devour  "  certain  sluggish  unprotected  larvae." 

There  are  many  observations  which  show  that  earwigs  in  some 
localities  prey  upon  small  insect  larvae,  and  in  certain  instances 
they  have  been  recommended  as  a  means  of  diminishing  plant 
pests.  Thus  the  following  references,  as  also  others  quoted  in  this 
paper,  have  appeared  in  issues  of  The  Review  of  Applied  Entomo- 
logy, 1913—1918.  Bernard  {Technique  des  traitements  contre  les 
Insectes  de  la  Vigne,  Paris,  1914)  states  that  they  devour  the 
pupae  of  one  or  more  of  Clysia  amhiguella,  Polychrosis  botrana, 
and  Sparanothis  pilleriana  {v.  also  'Kirkaldy,  "^^i^o??*.  xxxiii. 
1900,  p.  87).  Dobrodeev  {Mem.  Bur.  Entom.  of  Gent  Board 
of  Land  Administration  and  Agric,  Petrograd,  XL  no.  5,  1915) 
makes  a  similar  report  as  regards  the  destruction  of  the  first  tAvo 
Tortricidae  named  above  by  earwigs.  Molz  {Zeits.  Angeiuandte 
Ghemie,  Leipzig,  xxvi.  nos.  77,  79,  1913,  pp.  533,  587)  speaks  of 
earwigs  as  natural  enemies  of  the  vine  moth.  Feytaud  {Bull. 
Soc.  Etude  Vidg.  Zool.  Agric.  Bordeaux,  xv.  nos.  1—8,  Jan.— Aug 
1916,  pp.  1,  21,  43,  52,  65,  88)  states  that  earwigs  destroy  the 
eggs  and  larvae  of  the  coccid  vine  pests  Eidecanium  persica  and 
(probably)  Pulvinaria  vitis.     Harrison  in  "An  unusual  parsnip 


hy  birds  of  the  Coiiiiiwn  Earwig  (Forficula  auricuLiria)     175 

pest"  {Entomologist,  XLVI.  Feb.  1913,  p.  59)  reports  them  as 
most  effective  in  killing  and  eating  Depressaria  heradicwa,  the 
"parsnip  web-worm."  Brittain  and  Gooderham  (Canad.  Entorn., 
London,  Ont.,  XLVii.  no.  2,  Feb.  1916,  p.  37)  make  a  similar  state- 
ment. 

There  is  no  doubt  that  our  knowledge  of  the  bionomics  of 
the  earwig  is  at  present  very  imperfect.  As  in  the  case  of  other 
very  common  animals  far  too  much  has  been  taken  for  granted. 
The  earwig's  nocturnal  habit,  its  tendency  to  assemble  in  great 
numbers  between  two  closely  apposed  surfaces,  and  its  "frightening 
attitude  "  of  flexing  its  abdomen  dorsalwards  with  opened  forcipes 
all  tend  to  give  it  a  reputation  for  evil  which  very  probably  is 
but  partially  deserved.  We  all  know  how  the  habit  of  entering 
crevices  is  responsible  for  the  belief  that  it  gnaws  through  the 
tympanic  membrane  with  the  result  of  mania  or  even  death. 
Perce-oreille  speaks  for  itself.  It  seems  fairly  established  that 
its  universally  bad  reputation  among  gardeners  is  founded  on 
tradition  and  want  of  judgment  combined  with  neglect  of  the 
increasing  evidence  that  its  presence  is  sometimes  beneficial  by  its 
destructiveness  to  more  harmful  insects  than  itself  That  it  eats 
the  petals  of  dahlias  and  chrysanthemums  to  some  extent  is  true, 
but  as  far  as  my  own  observations  go  the  outlay  of  time  and 
material  devoted  to  the  traditional  protection  of  the  flowers  by 
inverted  flower  pots  stuffed  with  straw  seems  hardly  worth  while. 
The  great  attraction  which  the  flowers  have  for  earwigs  seems  to 
be  the  closeness  and  number  of  their  petals,  which  provide  a 
daytime  shelter  whence  nightly  excursions  for  feeding  are  made. 
Anyone  possessing  a  garden  may  greatly  add  to  our  knowledge  of 
favourite  foods;  observation  at  night  is  particularly  needed.  As 
regards  garden  varieties  of  roses  the  case  against  earwigs  is 
probably  more  severe. 

(c)     Capture  by  birds. 

During  the  last  decade  systematic  investigation  of  the  contents 
of  the  alimentary  canal  of  British  wild  birds  by  several  observers 
has  resulted  in  most  useful  information  as  to  which  should  be 
regarded  as  harmful  and  which  as  neutral  or  beneficial  to  agri- 
culture. It  is  manifest  from  the  laborious  and  painstaking  work 
now  at  our  disposal  that  many  of  the  reputations,  good  or  evil, 
which  certain  common  birds  have  in  the  eyes  of  farmers  and 
gardeners  need  considerable  revision,  in  some  cases  even  reversal. 

As  regards  the  capture  of  earwigs  by  birds,  it  appears  that 
they  are  not  a  favourite  food  when  we  bear  in  mind  how  numerous 
they  are  sometimes  and  that  they  are  large  enough  to  be  easily 
seized.  No  doubt  their  nocturnal  habit  affords  much  protection 
from  capture. 


176     Mr  Brindley,  Notes  on  certain  parasites,  food,  and  caj^ture 

Collinge  in  "  The  Food  of  some  British  Wild  Birds  "  (London, 
1913)  reports  on  the  contents  of  the  crop,  etc.,  of  29  of  the  com- 
monest species,  among  which  only  four  contained  earwigs,  and 
these  were  very  few  in  number.  Thus  in  404  House  Sparrows 
2  earwigs  were  found,  1  in  each  of  2  birds;  in  721  Rooks  2  ear- 
wigs were  found,  1  in  each  of  2  birds ;  in  40  Skylarks  3  earwigs 
were  found  among  2  birds;  in  64  Song  Thrushes  7  earwigs  were 
found  among  2  birds. 

Newstead  in  "  The  Food  of  some  British  Birds  "  (Sapp.  to  Journ. 
of  Board  of  Agric.  no.  9,  Dec.  1908)  records  observations  on  the 
swallowed  food  of  128  species,  the  outcome  of  871  post-mortem  and 
pellet  examinations  carried  out  in  various  years  from  1894  to  1908. 
He  finds  that  10  sj)ecies  had  eaten  earwigs,  the  numbers  of  birds 
examined  and  the  numbers  of  earwigs  found  being :  1  Whimbrel, 
40  earwigs ;  2  Green  Woodpeckers,  24  earwigs ;  2  Starlings,  3  ear- 
wigs; 1  Nuthatch,  3  earwigs;  1  Chaffinch,  1  Great  Titmouse, 
1  Redbreast,  1  Song  Thrush,  1  Whinchat,  1  Woodcock,  1  earwig 
each. 

Theobald  and  McGowan  in  '•'  The  Food  of  the  Rook,  Chaffinch 
and  Starling"  {Sapix  to  Journ.  of  Board  of  Agric.  no.  15,  May 
1916)  put  on  record  a  particularly  valuable  and  interesting  series 
of  observations,  as  they  examined  the  food  month  by  month 
during  nearly  2^  years,  viz.,  from  Jan.  1912  to  May  1914,  the 
inquiry  covering  277  Rooks,  748  Starlings,  and  527  Chaffinches. 
An  analysis  of  their  results  as  regards  earwigs  for  the  2^  years  is 
as  follows: 


3K 

^c^ 


Birds 
examined 

Earwigs 
found 

Average  number  of 

earwigs  taken  by 

each  bird 

Starling 

Chaffinch 

Rook 

372 

277 
121 

154 
7 
3 

•41 

•025 
•025 

Starling 

Chaffinch 

Rook 

376 

248 
156 

199 
5 
3 

■   -53 

•020 
•019 

I  have  divided  the  year  into  two  j)eriods  of  six  months  con- 
formably with  the  seasonal  presence  or  absence  of  earwigs  on  the 
surface  of  the  ground.    From  October  to  March  most  male  earwigs 


hy  birds  of  the  Common  Earwig  (Forficula  auriciilaria)     177 

die  and  the  females  are  hibernating.  In  view  of  this  it  is  curions 
that  earwigs  should  be  taken  as  numerously  during  this  period 
as  during  the  six  months  when  both  nymphs  and  adults  can  be 
found  easily.  The  numbers  recorded  for  Rook  and  Chaffinch  are 
small,  though  a  large  number  of  birds  were  examined.  The  Starling 
is  a  great  insect  eater;  is  it  possible  that  it  habitually  searches 
for  buried  insects  during  the  colder  months  and  devours  earwigs 
found  with  the  rest  ?  This  action  may  be  true  for  the  other  two 
birds  also.      The  figures  for  all  three  are  certainly  curious. 

So  we  find  only  13  species  of  birds  reported  as  having  captured 
earwigs,  and  most  of  them  as  very  sparingly.  The  Starling  is  not 
recorded  by  Collinge  as  an  earwig  eater. 

The  above  quoted  reports  certainly  suggest  that  wild  birds 
cannot  be  relierl  upon  to  diminish  earwigs  in  a  garden.  Many 
of  the  most  insectivorous  are  not  reported  as  feeding  upon 
earwigs  at  all.  They  may  be  distasteful,  and  a  large  number 
together  emit  a  well-defined  odour,  and  the  same  is  true  of  a 
number  preserved  in  alcohol.  Be  this  as  it  may,  domestic  fowls 
always  eat  them  readily,  a  fact  which  is  noted  by  Jones  {op.  cit.) 
in  the  case  of  the  invasion  of  Newport,  R.I.  He  also  mentions 
that  toads  will  eat  them. 

Miss  Maud  D.  Haviland,  Hon.  Mem.  B.O.U.,  to  whom  I  am 
indebted  for  assistance  with  regard  to  the  literature  of  the  subject 
and  for  kind  advice  in  the  preparation  of  these  notes,  informs  me 
that  she  has  noticed  a  Redbreast  take  earwigs  in  preference  to 
earthworms. 


ADDENDA. 

Under  (b). 
Mr  H.  Ling  Roth  informs  me  that  he  has  found  earwigs  very  destructive  to  iris 
pods,  with  resulting  premature  fall  of  seeds,  in  a  garden  at  Halifax,  Yorks. 

Under  (c). 
Gurney,  in  "Ornithological  Notes  from  Norfolk  for  1916"  {British  Birds,  x.  1917, 
p.  242j,  records  that  his  father  in  October,  1843,  found  several  earwigs  in  a  Stone 
Curlew. 


VOL.   XIX.   PART  IV. 


13 


178     M(ijo7^  MacMahov  and  Mr  Darling,  Reciprocal  Relations 


Reciprocal  Relations  in  the  Theorij  of  Integral  Equations.     By 
Major  P.  A.  MacMahon  and  H.  B.  C.  Darling. 


[Received  1  February  1918.     Read  4  February  1918.] 
1.     Let  f{oc)K{ajt)dx  =  ylr,{t) 

J  a, 

and  f2{cc)/c(a;t)da;  =  yjr^{t); 

J  a., 

then,  if  we  suppose  the  functions  f^,f  and  k  to  be  such  that  the 
order  of  integration  is  indifferent,  we  have 

fbi  rbo  rb, 

/     /i  (•^)  fa  (^t)  da;  =         dy  \     f  {x)/.,  {y)  k  {xyt)  dx 

=  \\Uy)i^i{yt)dy, 

or,  as  it  may  be  written, 

/     A(oo)yjr,(xt)dx=       f,(x)yfr,(xt)dx     (1). 

*i  J  a.2 

In  the  Messenger  of  Mathematics,  May  1914,  p.  13  Mr  Rama- 
nujan  has  employed  this  result  to  deduce  a  number  of '  interesting 
relations  between  definite  integrals.  The  method  is  very  suggestive 
and  appears  capable  of  considerable  extension.     For  example,  if 

f{x)K[e{x,t)\dx  =  ^lr,{t)\ 

[b.  \  (2), 

and  /    fM'c{e{x,t)]dx  =  ^lrM 

*^^n         \j^  (•^)  ts  [0  {a;  01  dx  =  ^J,  {x)  f,  {0  (x,  t)}  dx    . .  .(3), 

provided  that  0  {x,  6  {y,  t)]=- d  [y,  0{x,  t)\ (4). 

The  functional  equation  (4)  is  satisfied  by 

0{^,t)  =  cl,-^f(x)  +  cl>(t)\ (5), 

where  /  and  0  are  arbitrary  functions ;  which  is  a  general  form  of 
solution  and  includes  among  others  such  solutions  as 

H^>t)  =  c}>-^{f(x).cl>{t)} (6), 

^     ^     \f(^)  +  cP{t)\     ^'>' 

^{^,t)  =  cf^~^f(x)  +  cf,{t)+f(x)cf,(t)]    (8). 


in  the  Tlieory  of  Integral  Equations 


179 


Thus,  to  derive  (7)  from  (5)  let 

f{x)  =  coth-i  [P{oc)],     (/>  (0  =  coth-i  [(^1  {t)\  : 

then  (5)  becomes 

0-'  [coth-'  [F{x)]  +  coth-i  {(^1  {t)  W 

Now  let  (^"^  (^)  =  u, 

then  ir  =  <^(m)  =  coth"' 1^1  («)}, 

whence  ^i  {u)  =  coth  0, 

and  u  =  4>r^  (coth  2) ; 

that  is  (/)-'  (^)  =  01-1  (coth  ^), 

and  therefore  (5)  reduces  to 

_^\F{x)4>,(t)+l] 
F{x)  +  (ji,(t) 


0r 

which  is  of  the  form  (7 ). 

As  an  example  of  the  use  of  (2)  and  (3)  in  the  determination 
of  relations  between  integrals,  let 

/i  (•'^)  =  sin  X,    /,  {x)  =  cos  X, 

and,  using  the  form  (6)  for  0,  let 

0{x,  «)  =  e»'-'o.'^', 

K  (x)  =  X. 

bi  =  b.2  =  a,     Ui  =  «y  =  0, 


and 

Then,  putting 

we  have  from  (2)         yfr^  (t)  =       sin  x .  e^iog'' 

J  0 


dx 


(log  t .  sin  a  —  cos  a)  e^^^st  4. 1 

^  l  +  (logO'  ~ 


and 


yfr..it)=\    cos.'r.e-'"'°s'rf.« 

Jo 

(log  t .  cos  a  +  sin  a)  e"^^^  —  log  ^ 


l+(logO-^ 

Substituting  these  values  in  (3),  and  then  putting  log^  =  l/r 
for  brevity,  we  obtain 

'*"  [x  sin  {x  —  a)  +  7'  cos  {a;  —  a)}  e'**'"'  , 


r'  +  «' 

' -'  it'  sin  X  -\-  r  cos  x 


i  ■-'  X  SI 
•'0 


y.2   _j_    ,^2 


f/./ 


=  0: 


13—: 


180     Major  MacMalion  and  Mr  Darling,  Reciprocal  Relations 
so  that,  provided  r  is  not  zero,  we  have 


X  sin  {x  —  a)  +  r  cos  {x  —  a)\  e"'^'^ 


J  0  r-  +  x^ 


X  sm  X  -{-r  cos  .r  , 

r^  +  ic-  ^ 

an  identity  which  may  be  verified  by  differentiation  with  respect 
to  a.     Putting  x  =  r  tan  ^  and  then  replacing  ^  by  a;,  (9)  becomes 


•*^'^~'  «/'•  cos(^  +  a-  r  tan  .^■)    „  tan  x 


'^-^"^'^    '    "^ ■_::^-J  g  a  tan  x  ^^ 

J  0  cos  X 


.tan  1  a/;-  ^^g  (^  _  ^  ^^^  ^A 

=  ^ ax (10), 

J  0  cos  X 

which  admits  of  ready  verification  by  differentiation  with  respect 
to  a.  The  identities  (9)  and  (10)  hold  generally,  provided  that 
the  constants  are  finite;  we  have  seen  that  r  must  not  be  zero.  It 
will  be  noticed  that  both  (9)  and  (10)  are  of  the  form 

Jo  Jo 

where  the  upper  limits  of  integration  involve  a. 

2.     As  another  illustration  of  how  the  method  admits  of  genera- 
lisation, let 

fAx)'c{0{x,t)]dx  =  y\r,{t). 
J  «, 

rb, 
and  f2{x)K{d  {x,  t)]  dx  =  yjr,  (t) : 

J  0.2 

fbi  .     fb, 

then         I     /i  (x)  v/^a  {\  (x,  t)}  dx=\     fo  {x)  f,  {\  (x,  t)}  dx 

J  a,  J  a, 

when  \  {x,  t)  =  4>^-^  {/(x)  +  (j>,  (t)} 

and  e(x,t)=g{f(x)  +  cl>,it)}, 

f,  g,  (pi  and  (f>2  being  any  functions.  It  should  be  observed  that  A. 
becomes  6  when  (f)^  =  ^2  and  g  =  02~^.  Other  corresponding  pairs 
of  functions  are 

\(^,O  =  </>rM/(^')-0i(O), 

0(x,t)^g{f(x).cj>,{t)}> 

and  M^^0  =  <^r^J4^r\^4^|, 

'f(x)cf>,{t)  +  l] 


e{x,t)=g 


f{x)  +  (f>,{t) 


so  that 


in  tJie  Theory  of  Integral  Equations  181 

8.     A  further  extension  is  obtained  when  the  kernel  k  includes 
more  than  one  parameter  t;  thus  let 

/i  (x)  K  [6  {x,  ti ,  Q}  dx  =  -f,  (^1 ,  t,), 
/„  (x)  K  [d  {x,  t„  t^]  dx  =  yfr.  (t, ,  t.^, 

J  a.2 

\    fi  (!/)  «  [^  {!/>  f^  (^>  ii>  Q>  V  {x,  ti ,  t)}]  di/ 

J  a, 

=  -v/tj  \/j,  {x,  ti,  t^,  V  {x,  ti,  t.^\ 

and 

f  Vi  (//)  «  [^  y^  f^  (•'•'  ^i>  Q,  V  {x,  t, ,  f,)}]  dy 

=  \/ro  [^  {x,  t,,  ti),  V  {X,  t,,  Q). 

Now  consider 

/i  (•^-'O  -^/^a  [/A  (*',  ^1,  t;),  V  {x,  ti,  t^)}  dx 

=  f  V"i  (^)  (  I ' /3  (i/)/^  [0  [y,  ti  (,*•,  ^x,  t^},  V  {x,  t, ,  t,)}]  dy)  dx. 

■J   tti  ^  ■     fl2  ' 

This  double  integral  is  equal  to 

if      ^  {y/,  /x  (.r,  t„  t,),  V  {x,  t„  t.^}  =  e  [x,  iM  (y,  t„  t.;),  V  (y,  t„  L)}. 
Now  suppose 

fl  (X,  t„   «,)  =  <^,-'  [f{x)  +  (/>!  {t,)  +  (/>!  (^2)}. 

^  {x,  t„  L)  =  </)-!  \2f(x)  -f-  (/>!  (^0  +  </), (t,)} ; 
then     6^  {y,  /j.{x,  ti,  t,),  v{x,  t^,  t,)} 

=  </>-!  {2/(2/)  +  2/  (*■)  +  (/),  (t,)  +  01  (t,)  +  (/),  (^0  +  0.  (g). 
This  is  symmetrical  in  x  and  y,  so  that  we  may  write 
/^(^■,  ^1,  ^2)  =  0rM/3(*')+ 03(^1,  4)j, 
/.  C^-,  t„  t,)  =  0,-^  1/4 (^0  +  04  (^1,  ^2)}, 
0  (a^,  t„  t^  =  g  [f,  (x)  +f,  (x)  +  01  {t,)  +  0,  (QK 
leading  to 

5'{/3(^)+/4(i/)+/;(*')+/4(^O+03(^l,    ^  +  04(^1,    t.^\, 


182     Major  MacMahon  and  Mr  Darling,  Reciprocal  Relations 
which  is  symmetrical  in  x  and  v/ ;  and  hence  it  follows  that 
I    J[(x)ylrn  {/m(x,  ti,  L),  v{x.  t^,  ig)}  dx 

■  '  a, 

=  I     fo,{x)'\^i  {/"-(*■>  t\,  ^2))  v{^>  ^1)  ^2))  dx. 
As  a  particular  case  we  may  write 

^L  {x,  t, ,  L)  =  cf^r'  (./X^O  +  ^1  (A)  +  ^1  (^2)}, 

V  (X,    t,  ,   t.;)  =   (/),-'    {/(*■)  +  02  (^1)  +  C^2  (4)}, 

0  (x,  t„  Q  =  g  {2/(*0  +  4>,  (t,)  +  4>,  (01, 

and  again 

^l{x,  t„  Q  =  ct>-'  {^,f{x)  +  (i>(Q  +  cj)(t,)], 

0  {x,  t„  Q  -  ct>-'  {/(x)  +  0  (^0  +  (/)  (t,)}, 

the  case  where  /x  i^  y  and  each  resembles  0  as  much  as  possible. 
It  is  evident  that  the  case  in  which  the  kernel  includes  any  number 
of  parameters  may  be  treated  in  the  same  manner  and  presents 
little  difficulty. 

4.     The  method  may  also  be   extended  to  double   integrals. 
Thus  let 

/i  {^>  y)  K^  [^  (*'.  y>  ii>  4)1  dxdy  =  f^  (t,,  to), 

J  »!   J  a,' 

/■2  (^-^  y) «  [^  (^,  2/.  ii,  4)1  dxdy  =  -f .  (^i,  4) ; 

[b,      rb,' 

then  /     /i  {cc,  y)  i/r^  {/x  (^■,  y,  t^,  4),  ^  (a-',  y,  4,  4)}  c^^'f^^/ 

J  «,    >/  a,' 


6/ 


/2(^,  2/)'fi  l/^(«>  y.  4,  4)>  ^(^S  ^»  4,  4)1  dxdy 

if  ^  {^r,  w,  yu,  (a;,  y,  ^j,  4),  v  {x,  y,  t^,  t^)] 

=  0  [x,  y,  IX  {z,  w,  ti,  t.^,  V  (z,  IV,  ti,  4)1- 
If  A,  B,  G,  D,  E  be  functional  symbols,  one  solution  is 
ix{x,  y,  t„  O  =  A-'  [B{x,  y)  +  C {t„  t,)] 
v{x,  y,  t„  t,)  =  D-^[B{x,  y)  +  E(t„  t,)} 
0  (x,  y,  t„  t,)  =  B  (x,  y)  +  kA  (t,)  +  ^D  (t,). 


in  the  Theory  of  Integral  Equations  183 

5.     Let  US  next  consider  the  case  of  three  integral  equations 

!'\f\{x)K{dU;t)}dx  =  f,{t), 
J  «, 

/;  (a:)  K  [e  {x,  t)]  dx  =  ^/r,  (t), 


J  a,, 

r\f,(x)K{d(x,t)]dx=f,(t). 


We  have 

'yA''^)f.{OOr,t)\ir,{d{x,t)}dx\ 

=  r  Mx)^}r,{e{x,  t)}f,{e(x,  t)]  dx\ (11), 

=  fV;cr)  ti  {^(^'>  01  ir,{0(x,  t)}  dx  ] 

if  certain  conditions  are  satisfied.     For 

'"'  f\{x)y^,[e{x,t)]ylr,[e{x^t)]dx 

=  I '' ./;  i-'^)  f '  /.  (z/)  '^  [^  (>/>  t)\  dy  f  V;  {z)  K\e{z,  t)\  dzdx, 

and  the  equalities  (11)  will  hold  good  if,  for  example,  k  (x)  =  x'^  and 

d[y,d{x,t)\.e[z,e{x,t)] 

is  unaltered  by  the  circular  substitution  {xyz). 

Now  suppose  that  ^ 

e{x,t)^f{x)t-^    (12;; 

then  0  [y,  0  {x,  t)] .  6  {z,  d  (x,  t)\  =f(y)f(z)  0  {x,  t) 

Hence  if  k(x)  —  x'^  the  relation  (12)  satisfies  the  conditions.  The 
generalisation  to  the  equality  of  n  integrals  is  apparent,  and  in 
that  case 

0(x,  t)=f{x)t''-'^ 
is  a  solution. 
We  have  also 

/i  (.'•)  f;  1^  (*•,  0}  ^3  {^  (.'<-■,  t)]  dx 


fb, 
=        f,  (.'/;)  yjfs  {\  (x,  t)}  ^fr,  {X  (x,  t)]  dx 

J  rta 


184  Major  MacMahon  and  Mr  Darlwcf 

if  X  {x,  t)  =f{x)  t'\     e  (x,  t)  =  { f{x)Y'-'t''', 

and  in  particular  if 

X  {x,  t)  =f{x)  t^,     e  {x,  t)  =  [\  {x,  t)Y-''. 
A  solution  may  also  be  obtained  when  k  (x)  =  [f,  in  which  case 

K[e[y,e{x,t)\'\.K[d{z,  6'(^,  0}]  =  e^'•^'^^''■'*^^'^^'''^^''•*^^• 
Putting  d{x,t)=f{a^  +  lt, 
we  have 

e  [y,  d  (x,  t)]  +  e{z,e  (x,  t)]  =/(y/)  +f(z)  +  f(x)  +  ^t, 
which  is  of  the  symmetrical  form  required. 

6.  In  the  cases  investigated  above  the  kernels  of  the  several 
integral  equations  have  been  functions  of  the  same  form.  It  is, 
however,  easy  to  extend  the  method  to  the  case  where  the  kernels 
are  functions  of  different  form.     Thus  if 

/i  (x)  /ci  {0  (x,  t)\  dx  =  -v/tj  (t), 

bo 
/a  (x)  K.  {6  (x,  t)}  dx  =  ylr.2  (t), 

we  are  led  to  the  condition 

K,  [6  [y,  \  (x,  t)]]  =  K,  [6  [x,  \  (y,  t)]]. 

Case  1.     Let  Ki(z)—z,  k2{1/z)  =  z;  then  the  condition  becomes 
0{y,X(x,t)].d{x,\(y,t)}^l; 
a  solution  of  which  is 

d(x,  t)^xWi^\  (A (OK %  {0(0,  F{^)i 

where  \  {x,  t)  =  (jr^  F  {x), 

and  ;j^  is  any  function. 

Case  2.     Let  k^  (z)  =  z,  k.,(-z)^z;  then  the  condition  is 
0{y,X(x,t)}  +  e{x,\{y,  01=0; 
a  solution  of  which  is 

d  (x,  t)  =  x  {F{x).,  c/,  (01  -x\<^  (0.  F{x)\. 
Case  3.     Let  k^  {z)  =  z\  k,_  (s)  =  (1  -  ^0'" ;  then  the  condition  is 

\0[y,\{x,t)\J^ld{xMy^m=^\ 

a  solution  of  which  is 

e {x,  t)  =  x[F{x),  </>(0} ^ lixWia^), 4>{i)]y  +  (%{</> (0, F{x)]yr- 


Prof.  Stanley  Gardiner  and  Prof.  Nuttall,  Fislt-freezing     185 


Fish-freeznuj.  By  Professor  Stanley  Gardiner  and  Professor 
Nuttall. 

[Read  18  February  1918.] 

Fish-freezing  commenced  in  1888,  in  connection  with  Western 
American  sahnon.  It  was  started  to  preserve  the  excess  of  fish 
caught  during  the  runs  for  canning  in  the  shick  season.  The  busi- 
ness proved  so  profitable  that  fish  began  to  be  distributed  all  over 
North  America  and  exported  to  Europe,  the  chief  market  in  the 
latter  being  Germany.  The  fish  are,  as  soon  as  possible  after  catch- 
ing, brought  to  the  refrigerator,  frozen  dry  on  trays  at  about  10°  F., 
this  process  taking  about  36  hours.  The  fish  then  are  drawn  into 
a  room  at  20"  F.,  where  they  are  dipped  into  fresh  watei',  their  sur- 
faces being  thus  covered  with  a  glaze  of  ice.  They  are  then  packed  in 
parchment  paper  in  strong  wooden  cases  and  exported  to  Europe 
by  refrigerator  cars  and  cold  storage  steamers.  The  process  is  also 
applied  to  halibut,  haddock,  cod,  pollack  and  various  flat  fish  in 
America.  It  succeeds  in  preserving  the  fish  for  an  indefinite  period 
of  time,  but  the  product  breaks  up  in  cooking,  tending  to  become 
rather  woolly  and  loses  flavour  and  aroma. 

To  meet  this  a  fresh  process  has  now  been  developed,  freezing 
the  fish  in  brine  consisting  of  about  18  per  cent,  of  salt  at  a  tem- 
perature of  5°  to  20"  F.  The  brine  is  an  excellent  conductor  of 
heat  and  cold.  A  large  fish  freezes  thoroughly  in  three  hours,  a 
herring  in  twenty  minutes.  After  freezing,  the  fish  returns  to  the 
same  condition  as  it  was  when  placed  into  the  brine;  there  is  no 
woolliness,  no  loss  of  flavour  or  aroma.  The  difference  is  due  to  the 
fact  that,  whereas  in  dry  freezing  there  is  a  breaking  up  of  the 
actual  muscular  fibres,  due  to  the  formation  of  ice  crystals,  in  brine 
freezing  the  ice  crystals  are  so  small  that  the  muscular  fibres  are 
entirely  unaffected  and  on  thawing  return  to  the  normal.  In  neither 
form  of  freezing  is  there  danger  from  moulds  or  putrefaction  if  the 
fish  is  stored  below  20^  F. 

The  authors  advocate  the  creation  of  a  vast  store  of  frozen  her- 
rings against  time  of  scarcity,  instead  of  the  herrings  being  pickled 
and  exported.  The  value  of  fish  as  food  is  weight  for  weight  about 
the  same  as  meat,  containing  the  same  constituents.  If  the  excess 
of  the  herring  catch  were  stored  in  this  way,  there  would  be,  on 
pre-war  figures,  a  store  of  herrings  in  this  country  to  meet  the 
necessity  for -albuminous  food  in  the  British  Isles  for  at  least  eight 
weeks. 


186     Mr  Sahni,  On  the  branching  of  the  Zygopteridean  Leaf,  etc. 


On  the  branching  of  the  Zygopteridean  Leaf,  and  its  relation  to 
the  probable  Pinna-nature  of  Gyropteris  sinuosa,  Goeppert.  By 
B.  Sahni,  M.A.,  Emmanuel  College.  (Communicated  by  Professor 
Seward.) 

[Read  20  May  1918.] 

( 1 )  The  supposed  quadriseriate  "  pinnae  "  of  forms  like  Staurop- 
teris  and  Metaclepsydi^opsis  are  tertiary  raches,  the  vascular  strands 
of  the  secondary  raches  (pinna-trace-bar,  Gordon)  being  completely 
embedded  in  the  cortex  of  the  primary  rachis.  All  Zygopterideae 
therefore  have  a  single  row  of  pinnae  on  each  side  of  the  leaf. 
(2)  This  revives  the  suggestion  that  Gyropteris  sinuosa  Goepp.  is 
a  free  secondary  rachis  of  a  form  like  Metaclepsydropsis.  (3)  The 
genus  Glepsydropsis  should  include  Ankyropteris  because:  a.  A 
fossil  described  in  1915  (Mrs  Osborn,  Brit.  Ass.  Rep.,  p.  727)  com- 
bines the  leaf-trace  of  Glepsydropsis  with  the  stem  of  Ankyropteris, 
the  leaf-trace  in  both  arising  as  a  closed  ring.  h.  In  G.  antiqua 
Ung.  also  the  leaf-trace  arose  similarly,  as  shown  by  a  section 
figured  by  Bertrand  {Progressus  1912,  fig.  21,  p.  228)  in  which  a 
row  of  small  tracheides  connecting  the  inner  ends  of  the  peripheral 
loops  represents  those  lining  the  ring  before  it  became  clepsydroid 
by  median  constriction. 


3 


The  Structure  o/Tmesipteris  Vieillardi i)aw^.  By  B. Sahni, M.A., 
Emmanuel  College.     (Communicated  by  Professor  Seward.) 

[Read  20  May  1918.] 

The  most  primitive  (least  reduced)  of  the  Psilotales.  Specifically 
distinct  from  T.  tannensis  in  (1)  erect  terrestrial  habit,  (2)  distinct 
vascular  supply  to  scale-leaves,  (3)  medullary  xylem  in  lower  part 
of  aerial  stem. 

On  Acmopyle,  a  Monotypic  New  Galedonian  Podocarp.  By 
B.  Sahni,  M.A.,  Emmanuel  College.  (Commimicated  by  Professor 
Seward.) 

[Read  20  May  1918.] 

Indistinguishable  ivova. Podocar pus  in  habit,  vegetative  anatomy, 
drupaceous  seed,  megaspore-membrane,  young  embryo,  male  cone, 
stamen,  two-winged  pollen  and  probably  male  gametophyte.  Chief 
differences:  (1)  seed  nearly  erect;  (2)  epimatium  nowhere  fi"ee  from 
integument,  even  partaking  in  formation  of  micropyle;  (3)  outer 
flesh  with  a  continuous  tracheal  mantle  covering  the  basal  two-thirds 
of  the  stone. 


PROCEEDINGS    AT   THE    MEETINGS    HELD   DURING 

THE   SESSION    1917—11)18. 

ANNUAL   GENERAL   MEETING. 
October  29,   1917. 
In  the  Comparative  Anatomy  Lecture  Room. 

Dr  Mark,  President,  in  the  Chair. 
The  following  were  elected  Officers  for  the  ensuing  year  : 
President: 
Dr  Marr. 

Vice-PresideiUs : 

Prof.  Newall. 
Dr  Doncaster. 
Mr  W.  H.  Mills. 

Treas'itrer  : 
Prof.   Hobson, 

tSecretaries  : 

Mr  A.   Wood. 
Mr  G.   H.  Hardy. 
Mr  H.  H.  Brindley. 

Other  Members  of  Council  : 

Dr  Bromwich. 

Mr  C.  G.   Lamb. 

Mr  J.  E.   Purvis. 

Dr  Shipley. 

Dr  Arber. 

Prof.  Bitfen. 

Mr  L.   A.  Borradaile. 

Mr  F.  F.  Blackman. 

Prof.   Sir  J.  Larmor. 

Prof.  Eddington. 

Dr  Marshall. 

The  following  Communications  were  made  to  the  Society  : 

1.  On  the  convergence  of  certain  multiple  series.    ByG.  H.Hardy, 
M.A.,  Trinity  College. 

2.  Bessel  functions  of  large  order.      By  G.   N.   Watson,  M.A., 
Trinity  College. 


188  Proceedings  at  the  Meetitujis 

3.  A  particular  case  of  a  theorem  of  Dirichlet.    By  H.  Todd,  B.A., 
Pembroke  College.     (Communicated  by  Mr  H.  T.  J.  Norton.) 

4.  On  Mr  Ramanujan's  Empirical  Expansions  of  Modular  Functions. 
By  L.  J.  MoRDELL.     (Communicated  by  Mr  G.  H.  Hardy.) 

5.  Extensions  of    Abel's    Theorem    and    its    converses.       By    Dr 
A.  KiENAST.     (Communicated  by  Mr  G.  H.  Hardy.) 


November  12,   1917. 
In  the  Comparative  Anatomy  Lecture  Koom. 

Professor  Marr,  President,  in  the  Chair. 

The  following  Communications  were  made  to  the  Society  : 

1.  Some  experiments  on  the  inheritance  of  weight  in  rabbits.  By 
Professor  Punnett  and  the  late  Major  P.  G.  Bailey. 

2.  The  Inheritance  of  Tight  and  Loose  Paleae  in  Avena  nuda 
crosses.  By  A.  St  Clair  Caporn.  (Communicated  by  Professor 
Bitten.) 


February  4,   1918. 
In  the  Comparative  Anatomy  Lecture  Room. 

Professor  Marr,  President,  in  the  Chair. 

The  following  Communications  were  made  to  the  Society: 

1.  On  certain  integral  equations.     By  Major  P.  A.  MacMahon. 

2.  (1)    Sir  George  Stokes  and  the  concept  of  uniform  convergence. 
(2)    Note  on  Mr  Ramanujan's  Paper  entitled  :  On  some  definite 

integrals. 

By  G.  H.  Hardy,  M.A.,  Trinity  College. 

3.  Asymptotic  expansions  of  hypergeometric  functions.    By  G.  N. 
Watson,  M.A.,  Trinity  College. 

4.  (1)    On  certain  trigonometrical  sums  and  their  applications  in 

the  theory  of  numbers. 
(2)    On  some  definite  integrals. 
By  S.    Ramanujan,    B.A.,    Trinity  College.      (Communicated  by  Mr 
G.  H.  Hardy.) 


Proceedings  at  the  Meetings  189 

February   18,    191S. 
In  the  Comparative  Anatomy  Lecture  Room. 

Professor  Marr,  President,  in  the  Chair. 

The  following  were  elected  Fellows  of  the  Society  : 

E.  Lindsay  Ince,  B.A.,  Trinity  College. 
S,  Ramanujan,  B.A.,  Trinity  College. 

The  following  Communications  were  made  to  the  Society  : 

1.  Fish-fi-eezing.     By  Professor  Stanley  Gardiner  and  Professor 

NUTTALL. 

2.  Shell    deposits    formed    by    the    flood    of   January    1918.     By 
P.  Lake,  M.A.,  St  John's  College. 

3.  (1)    Reactions  to  Stimuli  in  Corals. 

(2)    Is  the  Madreporarian  Skeleton  an  Extraprotoplasmic  Secre- 
tion of  the  Polyps  1 

By  G.  Matti^ai,  M.A.,  Emmanuel  College.   (Communicated  by  Professor 
Stanley  Gardiner.) 

4.  Notes  on  certain  parasites,  food,  and  capture  by  birds  of  Forficuhi 
cmricnlaria.     By  H.  H.  Brindley,  M.A.,  St  John's  College. 


May  20,   1918. 

In  the  Botany  School. 

Professor  Marr,  President,  in  the  Chair. 
The  following  was  elected  a  Fellow  of  the  Society  : 

C.  Stanley  Gibson,  Sidney  Sussex  College. 
The  following  Communications  were  made  to  the  Society  : 

1.  (1)   On  the  branching  of  the  Zygopteridean  Leaf,  and  its  relation 

to    the    probable    Pinna-nature    of    Gyropteris   simiosa, 
Goeppert. 

(2)  The  Structure  of  Tmesipteris  Vieillardi  Dang. 

(3)  On  Acmopyle,  a  Monotypic  New  Caledonian  Podocarp. 

By  B.  Sahni,  M.A.,  Emmanuel  College.     (Communicated  by  Professor 
Seward.) 

2.  Asymptotic   Satellites  in  the    problem  of   three    bodies.      By 
D.  Buchanan.    (Communicated  by  Professor  Baker.) 


CONTENTS. 

PAGE 

Extensions  of  Abel's  Theorem  and  its  converses.  By  Dr  A.  Kienast, 
Kusnacht,  Zurich,  Switzerland.  (Communicated  by  Mr  G.  H. 
Hardy) 129 

Sir  Oeorge  Stokes  and  the  concept  of  uniform  convergence.     By  G.  H. 

Hardy,  M.A.,  Trinity  CoUege 148 

Shell-deposits  formed  by  the  flood  of  Jamtary,  1918.     By  Philip  Lake, 

M.A.,  St  John's  College 157 

7s  the  Madreporarian  Skeleton  an  Extraprotoplasmic  Secretion  of  the 
Polyps?  By  G.  Matthai,  M.A.,  Emmanuel  College,  Cambridge. 
(Communicated  by  Professor  Stanley  Gardiner)       .        .         .        .160 

On  Reactions  to  Stimuli  in  Corals.  By  G.  Matthai,  M.A.,  Emmanuel 
College,  Cambridge.  (Communicated  by  Professor  Stanley  Gar- 
diner)         164 

Notes  on  certain  .parasites,  food,  and  captitre  by  birds  of 'the  Common 
Earwig  (Forficula  auricularia).  By  H.  H.  Brindlet,  M.A., 
St  John's  College 167 

Reciprocal  Relations  in  the  Theory  of  Integral  Equations.     By  Major 

P.  A.  MacMahon  and  H.  B.  C.  Darling 178 

Fish-freezing.    By  Professor  Stanley  Gardiner  and  Professor  Nuttall     185 

On  the  branching  of  the  Zygopteridean  Leaf,  and  its  relation  to  the  pro- 
bable Pinna-nature  o/Gyropteris  sinuosa,  Ooeppert.  By  B.  Sahni, 
M.A.,  Emmanuel  College.     (Communicated  by  Professor  Seward) .     186 

The  Structure  of  Tmesipteris  Vieillardi  Bang.     By  B.  Sahni,  M.A., 

Emmanuel  College.     (Communicated  by  Professor  Seward)     .        .     186 

On  Acmopyle,  a  Monotypic  New  Caledonian  Podocarp.    By  B.  Sahni, 

M.A.,  Emmanuel  College.     (Communicated  by  Professor  Seward)  .     186 

Proceedings  at  the  Meetings  held  during  the  Session  1917 — 1918   .         .     187 


(>.r 


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PROCEEDINGS 

OF  THE 

CAMBRIDGE  PHILOSOPHICAL 
SOCIETY 


VOL.   XIX.     PART   V. 

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e 


PROCEEDINGS 

OF   THE 


On  Certain  Trigonometrical  Series  which  have  a  Necessary  and 
Suffijcient  Condition  for  Uniform  Convergence.     By  A.  E.  Jolliffe. 

(Communicated  by  Mr  G.  H.  Hardy.) 

[Received  1  June  1918;  read  28  October  1918.] 

1.  The  series  S^n  sin  nd,  where  (a,i)  is  a  sequence  decreasing 
steadily  to  zero,  is  convergent  for  all  real  values  of  6,  and  it  has 
been  proved  by  Mr  T.  W.  Chaundy  and  myself*  that  the  series  is 
uniformly  convergent  throughout  any  interval  if  /?a,i-*-0,  this  con- 
dition being  necessary  as  well  as  sufficient. 

A  generalization  of  this  theorem  is  as  follows  : 

If  (Xn)  is  tt  sequence  increasing  steadily  to  infinity  and  (an)  is 
a  sequence  decreasing  steadily  to  zero,  then  the  necessary  and  suffi- 
cient condition  that  the  series  Sa„+i(cos  A,.,i^  — cosX,i_^i^)/^,  which  is 
coyiver gent  for  all  real  values  of  6,  shoidd  be  uniformly  convergent, 
throughoid  any  interval  of  values  of  6,  is  Xnan^b. 

I  shall  prove  rather  more  than  this,  viz.  that  the  condition  is 
sufficient  for  uniform  convergence  and  necessary  for  continuity. 

When  ^  =  0,  it  is  understood  that  the  value  assigned  to  any 
term  of  the  series  is  its  limit  as  6  tends  to  zero,  so  that  for  ^  =  0 
the  sum  of  the  series,  which  I  shall  denote  by  Sun,  is  zero.  Since, 
by  Abel's  lemma, 

i  Wn+i  +  ...  +  Up\<  2an+i/0, 

it  is  evident  that  there  is  continuity  and  uniform  convergence 
throughout  any  interval  which  does  not  include  ^  =  0,  so  that  it 
is  only  intervals  which  include  ^  =  0  that  we  have  to  consider. 

*  Proc.  London  Math.  Soc.  (2),  Vol.  15,  p.  214. 
VOL.  XIX.   PART  V.  14 


192     3Ir  Jollife,  On.  Certain  Trigonometrical  Series  which  have 

A  very  trifling  modification  of  the  analysis  which  follows  will 
show  that,  so  far  as  an  interval  which  includes  ^  =  0  is  concerned, 
the  same  results  hold  for  the  series 

2a„+i  (cos \n^  —  cos  \n+i^)  cosec  1)9, 

where  h  is  any  fixed  number.  If  either  |(X„+i  —  A,„)  or  ^(Xn+i  +  X„) 
is  always  an  integral  multiple  of  some  fixed  number  b,  then  X„. 
differs  by  a  constant  from  an  integral  multiple  of  26,  and  the  series 
is  periodic  with  a  period  Tr/b.  In  this  case  the  results  which  are 
true  for  an  interval  which  includes  ^  =  0  are  true  for  any  interval. 
The  particular  series  2a«sinn^  corresponds  to  b  =  ^,  X,,j=»  +^. 

2.  Since  the  sum  of  the  series  when  ^  =  0  is  zero,  it  follows 
that,  for  continuity  at  6  =  0,  the  sum  of  the  series,  when  6  is 
different  from  zero,  must  tend  to  zero  as  6  tends  to  zero  in  any 
manner.  In  particular,  the  sum  when  6  =  7r/2\„,  must  tend  to 
zero,  as  n  tends  to  infinity. 

When  6  =  7r/2\,j,  let  m  be  the  integer  such  that 

Xm-i^  $  TT  <  \n6- 

It  should  be  noticed  that  we  may  have  m  —  l  =  n,  and  that 

When  m—1  >n,  cos  \p-i6  —  cos \p6  is  positive,  so  long  as  p  is 
not  greater  than  ?7i  —  1,  and  consequently 

0{U^  +  Uo+  ...  +Um-i) 

>  a-n  (cos  Xi6  —  cos  \n^)  +  «w-i  (cos  X„^  -  cos  A,„,_i 0). 
Also,  by  Abel's  lemma, 

6  (Um  +  U,n+i  +  ...  +  U,n+g)  >  "m  (cOS  X^O  -  1) 

for  all  values  of  q. 

Hence  the  sum  of  the  series  is  greater  than 

[an  COS  \6  -  {a,n-i  -  a„i)  cos  Xn-iO  -  am]IO, 

which,  since  a,„,_i  ^  a^  and  cos  Xm-i  ^  is  negative,  is  greater  than 

(an  cos  X^d-  a„,)/6  =  2Xn (««  -  «,»,)/-"■  +  bn, 

where  bn  denotes  a„(l  —cosXi6)/d  and  consequentl}^  tends  to  zero 
as  n  tends  to  infinity. 

When  m  —  1  =  ?i,  we  can  divide  the  series  up  into 

(U^  +  U2+  ...  +  Un)  +  (Um  +  Um+i  +...), 

and,  noticing  that  cos  X^-i  6  =  0,  we  see  that  the  sum  is  greater 
than  (an  cos  Xj^  —  am)/ 6,  as  before. 


a  Necessary  and  Suficiejit  Condition /or  Uniform  Convergence  193 

Hence  the  sum  of  the  series,  when  6  =  7r/2X,i,  can  in  no  case 
tend  to  zero,  as  n  tends  to  infinity,  unless  X,i  (a,i  —  a,n)  -^  0. 

If  X)i  (an  —  a,n)  -*  0,  then,  given  any  positive  number  e,  we  can 
find  V  such  that  X,i  (a„ -«,„)<  e  for  n^v.  Denote  ni  by  (n,  1) 
and  let  (/i,  2)  be  the  integer  formed  from  (n,  1)  in  the  same  way 
that  {ii,  1)  is  formed  from  n,  and  so  on.     Then 

((n  -  «n,  1  <  e/\n ,  (/„,  i  -  ttn,  o  <  e/X^,  i , , 

for  n  ^  V,  and  by  addition 

Un  <  €  (1/X,„  +  l/Xn,i  +  ...  +  lAn,i>)  +  (hi,p. 

Now  X,i_i  >  2\,i,,  \,i,^2  >  2X„_i,  and  so  on,  so  that  «„  <  2e/\„  +  cin,p. 
Also  when  %  is  fixed  we  can  choose  p  so  that  an^p  <  e/Xn,  and  we 
shall  have  therefore 

Xnttu  <  Be  (n  ^  v). 

Hence  XnCin^O  is  a  necessary  condition  that  the  sum  of  the 
series  should  be  continuous  at  ^  =  0,  and  a  fortiori  that  it  should 
be  continuous  throughout  any  interval  which  includes  ^  =  0. 

3.  To  show  that  this  condition  is  sufficient  for  uniform  con- 
vergence in  any  interval,  and  d  fortiori  for  continuity  at  any  point, 
it  is  sufficient  to  show  that 

I  Un+i+  ...  +Up\  <AM, 

for  all  values  of  6,  where  A  is  some  fixed  number  and  M  is  the 
greatest  value  of  Xt-a^  for  r^n  +  1. 

Since  the  value  of  the  series  is  changed  in  sign  only  by  changing 
the  sign  of  6,  it  is  sufficient  to  consider  positive  values  of  6  only. 
By  Abel's  lemma 

I  Un+i  +  ...  +Up\<  2an+i/d  <  2Xn+ian+i/7r, 

if  ^  ^  tt/Xji+j.    If  ^  ^  T^/Xp,  every  term  of  Un+i  +  . . .  +  Up  is  positive ; 
and,  if  u,.  is  one  of  these  terms, 

Ur  ^  M  (cos  Xr-iO  —  COS  X,.^)/X,.^ 

^  2if  sin  l(Xr  -  X,_i)  6  sin  h{X,  +  X,_,)  O/X^d  <  MO  {Xr  -  X,_i), 
so  that  Un+\  +  . . .  +  Up  <  MOXp  <  ttM. 

If    irjXp  <  6  <  7r/X„i+i,     let    Tr/Xq+i  <  6  ^  7r/Xg,    and     divide 

lln+i  +  ...  ^Up   up  into   Un+i  +  . . .  Uq  and  Uq+i  +  ...  +Up. 

Then  |  Un+i  +  ...  +  Uq\<  ttM,  and 

I  Uq+i  +  ...  +Up\<  2aq+^l0  <  2aq+iXq^,/7r  <  2ilf/7r. 

Therefore  j  Un+i  +  ...  -\- tip\  <  {ir  +  2/7r)  M. 

U~2 


194     Mr  Jolliffe,  On  Certain  Trigonometrical  Series  which  have 

Hence  for  all  values  of  6 

I  Un+^  +  ...  +  ?i^  i  <  (tt  +  2/7r)  M, 

and  therefore  the  condition  X,ia„  -*-  0  is  sufficient  for  uniform  con-"" 
vergence  and  a  fortiori  for  continuity  in  any  interval. 


il 


4.  If  'Xn  tends  to  infinity  more  rapidly  than  n,  the  series  does 
not  seem  to  be  capable  of  any  modification.  If  A,^  =  ^?2  +  B,  where 
A  and  B  are  fixed,  we  obtain  practically  the  series  2a„  sin  nO  and 
nothing  more.  But  when  X^  tends  to  infinity  more  slowly  than  n, 
and  with  a  certain  measure  of  regularity,  the  theorem  can  be 
transformed  in  an  interesting  manner.  We  have,  in  fact,  the 
following  theorem : 

If  \n  tends  steadily  to  infinity  and  \n+i  —  ^n  tends  steadily  to 
zero,  then  the  necessary  and  sufficient  condition  for  the  uniform 
convergence  of 

Zft^i  (Xji+i  ~  Xji)  sm  A.,jC7 
is  Xndn  -*  0. 

As  before,  I  prove  rather  more,  viz.  that  the  condition  is  suffi- 
cient for  uniform  convergence  and  necessary  for  continuity. 

This  theorem  will  follow  at  once  from  the  theorem  just  proved, 
if  we  can  show  that  the  series 

Sa«  {(cos  \nd  -  cos  Xn+i0)/6  -  (Xn+i  -  Xn)  sin  XnO] 

is  uniformly  convergent  throughout  any  interval.     Here  the  con- 
dition Xn+ittn  ■-*  0  is  equivalent  to  X^an  -^  0,  since  Xn+i  —  X^  -^  0. 
We  can  verify  immediately  that 

cos  y  —  cos  X  —  sin  y  sin  (sc  —  y) 

=  sin^ h{x  —  y)  (cos  y  —  cos  x)  +  ^  sin  {x  —  y)  (sin  x  —  sin  y). 

It  follows  by  Abel's  lemma  that,  if  Xn+i  —  X^  decreases  steadily, 
so  that  sm{Xn+i  —  Xn)d  and  sin  ^(Xn+i  —  Xn)  0  decrease  steadily, 
then 

S    {cos  Xn  0  —  cos  X,i+i 6  —  sin  {Xn+^  —  Xn)  0  sin  Xnd] 


-rt+1 


<  2  sin^  1  {Xn+i  ~Xn)6  +  sin  {Xn+i  -  X«) 0- 


Also,  given  any  e,  we  can  choose  v  so  that  Xn+i  —  X^  <  e  for  n  ^  v. 
Hence,  for  n'^v,  we  have 

p 

2  {cos  XnO  —  cos  Xn+i6  —  sin  (Xn+i  —  Xn)  6  sin  XnO] 

w+1 

<  2e2^2+66'<3e^, 
for  any  interval  of  values  of  0,  if  e  is  sufficiently  small. 


a  Necessary  and  Sufficient  Condition  for  Uniform  Convergence  195 

It  follows  also  that 

i   -&  I 

I  Z  sin  (X^+i  -  \,)6  sin  X,^^    <  2  +  Se^  <  3, 

|w  +  l  I 

for  n  ^  V.     Now 

(X„+i  -  Xn)  6  cosec  (X.,1+1  -  X„)  ^  -  1 
decreases  steadily  to  zero,  and  is  less  than 

Therefore 


2  6{\n+i  -  \i)  sin  Xu^  —  2  sin  (X„+i  —  X„)^  sin  XnO 

n+l  jj+1 

Hence 


<  e'6' 


p 

%  {(cos  X,i^  -  cos  \n+i6)ld  -  (Xn+i  -  ^/i)  sin  \nd} 

n+l 

<  3e  +  e^O  <  4e    (n  ^  i^). 

Hence  the  series 

2o„  {(cos  Xn^  -  cos  \n+id)l6  -  (X„+i  -  X„)  sin  X„^} 

is  uniformly  convergent  throughout  any  interval,  and  hence  the 
result  enunciated  follows. 

5.  If  instead  of  a  sequence  (X„)  we  have  a  function  X  (x)  such 
that,  as  iT-^oo,  X{a;)  increases  steadily  to  infinity  and  \'{a;)  de- 
creases steadily  to  zero,  then  Xn+i  —  ^n  decreases  steadily  to  zero. 
The  series  2  (X'„  —  X^+i  +  X,i),  where  \'n  denotes  the  value  of  X'  (x) 
when  x  =  n,  is  convergent  and  is  moreover  absolutely  convergent, 
since  X',^  —  \n+i  +  Xn  is  positive.  Hence,  by  Weierstrass'  M  test*, 
the  series  Sa,i  (X'„  —  X,i+i  +  X,i)  sin  X„^  is  uniformly  convergent 
throughout  every  interval.  It  follows  then  that  ajj,X„  ^  0  is  the 
necessary  and  sufficient  condition  that  the  series  Sa^X'„  sinX,j^ 
should  be  continuous  at  every  point  and  uniformly  convergent 
throughout  every  interval. 

In  particular  the  series  2a,i?i'~^  sin  (n'^),  where  t  is  any  real 
number  not  exceeding  1,  is  continuous  at  every  point  and  uniformly 
convergent  throughout  every  interval  if  n^a^-^O,  this  condition 
being  necessary  as  well  as  sufficient. 

*  Bi'omwich,  Infinite  series,  p.  113. 


196  Mr  TurnbuU,  Some  Geometrical  Interpretations 


Some  Geometrical  Interpretations  of  the  Concomitants  of  Tivo 
Quadrics.    By  H.  W.  Turnbull,  M.A. 

(Communicated  by  Mr  G.  H.  Hardy.) 

[Received  6  July  1918;  read  28  October  1918.] 

§  1.  In  the  Mathematische  Annaleii,  Vol.  LVi,  Gordan  has  given 
a  system  of  580  invariants  for  two  quaternary  quadratics.  It  appears 
that  by  carrying  out  the  processes  of  reduction  a  little  further,  the 
irreducible  forms  can  be  shewn  to  number  123  at  most.  That  is  to 
say,  the  system  is  about  as  complicated  as  the  ternary  system  for 
three  conies  which  Ciamberlini*  first  established.  It  is  therefore 
worth  while  to  give  geometrical  interpretations  to  members  of  the 
system  for  two  quadratics.  In  the  following  pages  about  a  hundred 
of  them  are  shewn.  The  geometrical  significance  of  the  residue 
appears  to  be  remote. 

Using  the  classification  introduced  by  Gordan,  the  numbers  of 
forms  of  each  type  J  which  have  not  been  reduced  are  shewn  in 
the  subjoined  Table.  The  rows  of  the  Table  give  the  numbers  of 
forms  of  each  particular  order  in  the  three  sets  of  coordinates  x, 
p,  u,  which  define  points,  straight  lines,  and  planes  respectively. 
Detailed  lists  of  these  forms  will  be  found  at  the  heads  of  the 
paragraphs  which  deal  with  separate  types. 


References 

Order  in  x,  p,  u 

J' 

J2 

Type 

J3 

J' 

J5 

Total  1 

i 
1 

§5 

Invariants 

5 

1 
5     1 

§6 

Covariants 

4 

1 

5     ! 

5) 

Con  travari  ants 

4 

1 

5     1 

§§  7-14 

Complexes 

6 

1 

1 

4 

4 

16     ' 

§  15 

Mixed  (1,  0,  1) 

1 

2 

3 

§21 

(1,  0,  3) 

4 

4     1 

5) 

(3,  0,  1) 

4 

4     i 

§22 

(2,  0,  2) 

1 

6 

7     ! 

§§  17-20 

(0,  1,  2) 

1 

6 

1 

1 

9    ! 

5) 

(2,  1,  0) 

1 

6 

1 

1 

9    ! 

§  18 

(0,  3,  2) 

1 

1    i 

)) 

(2,  3,  0) 

1 

1     ; 

§23 

(0,  2,  2) 

4 

4 

J5 

(2,  2,  0) 

4 

4 

§  16 

(1,  1,  1) 

4 

12 

16    ! 

§23 

(1,  2,  1) 

12 

6 

6 

24     1 

» 

(2,  1,  2) 

4 

4     1 

5) 

(3,  0,  3) 

2 

2 

Totals 

21 

7 

71 

12 

12 

123 

Ciamberlini,  Giornale  di  Matematiche,  Vol. 


of  the  Concomitants  of  Ttuo  Quadrics  197 

Notation. 

§  2.  Let  Ui,  Uo,  Ui,  u^  be  plane  coordinates;  and  let  v,  w  be 
cogredient  with  u.    We  may  then  typify  line  coordinates  by 

Pij  =  (uv)ij  =  UiVj  -  vflij,    (i,  j  =  1,  2,  3,  4)  ; 

and  X  or  point  coordinates  by  cc^  =  (uvtu)23^  and  three  similar  ex- 
pressions for  X2,  x-i,  ^4,  Then  the  symbolic  system  of  Gordan  can 
be  exhibited  as  follows. 

Let  the  point  equations  of  the  quadrics  be 

/=<''^' =  «/'=..., 
and  /"  =  h^-  =  bj^  =  .... 

Let  the  line  equations  be 

u=iApy  =  {A'py  =  ..., 

U'  =  {Bpy  =  (B'pf=.... 
Let  the  tangential  equations  be 

2  =  uj  =  uj  =  ..., 

X'  =  M,3"  =  ?'j3'^  =  .  .  .  . 

Then  the  connections  between  the  symbols  are 

A  =  (ta',    B  =  bb',    a  =  aa'a",    ^  =  bb'b". 
And  all  concomitants  of  the  system  can  be  expressed  in  terms  of 

TO  P'l'OT'^ 

4,    {dd'p),    (dd'd%),    {dd'd"d"'), 

where  d  signifies  a  or  b.  But  the  irreducibles  can  be  shewn  to  be 
composed  of  the  following  types, 

(la,  b^\  Ucc',  b^,  (Ap),  (Bp),  iia,  u^,  (abp),  (Abu),  {Ban),  (AB), 

a^,  ba,  (A/3x),  (Bax),  (a^p),  (AB)',  F„  F, ; 

where  (A^x)  —  a^a^'  —  a/a^  =  «^(V , 

say,  and  («/3/>)  =  UaVp  —  u^Va  =  luvp , 

{AB)'  =  (Ahu)k', 

F,  =  {(ibp)  a^'  -  (abp)  a^  =  (Abp^). 

Reciprocation. 

§  3.  By  interchanging  the  symbols  (a,  a),  (b,  /S),  (u,  x)  without 
altering  vl,  jB  or  ja,  we  obtain  from  any  given  concomitant  the 
reciprocal  form.    Thus  the  bracket  factors  (A^x)  and  (Abu)  are 


198  Mr  TurnhuU,  Some  Geometrical  Interpretations 

reciprocals.  So  also  would  be  (ahcu)  and  (a^ya;),  the  latter  being 
of  a  type  not  arising  for  less  than  three  quadrics.  Though  the 
process  by  which  Gordan  arrived  at  such  symbols  as  {A0iv)  and 
(a/3p)  was  purely  analytic,  it  is  interesting  to  observe  that  from 
the  geometrical  point  of  view  such  analytical  results  were  almost 
inevitable.  Below  will  be  found  several  examples  of  the  use  of 
this  principle  of  duality. 


The  fundamental  forms. 

§  4.  A  brief  investigation  would  reveal  the  importance  of  the 
following  forms,  to  which  special  symbols  are  therefore  attached. 

Let  /  denote      a^-,      f   denote      hx~, 

S  „  u^",      %'        „  %^ 

n  „  {Ap)\   W       „  {Bpf, 

k  „  {AjBxf,  k'        „  {Baxf, 

X  "  {Ahuf,   X         »  {Bauf, 

77^2       „  (abpf,  Dia       „  (ct^pT, 

and  C  „  {AB)(Ap)(Bp). 

Some  account  of  these  forms  may  be  found  in  Salmon,  Analytical 
Geometry  of  Three  Dimensions  (revised  by  Rogers),  Yol.  i,  Ch.  ix. 
There  %,  %,  %',  S'  are  denoted  by  a,  r,  r',  a'  (§  214) :  A^,  k'  are  the 
T,  r  of  §  215  ;  n,  7ri2,  O'  are  the  ^,  ^^,  ^'  of  §  217. 

Invariants. 

§  5.  The  irreducible  invariants  are  a^,  bj,  (ABf,  a^^,  bj  or 
the  A,  0,  ^,  @',  A'  of  Salmon,  §  200.  In  fact,  there  are  no  other 
types,  for  two  quadrics  of  any  dimension  n,  than  the  n  +  1  co- 
efficients of  X,  in  the  discriminant  of 

The  five  covariants  and  contravariants. 

I  6.  The  covariants  (w  +  1  in  the  case  of  w-ary  forms*)  are  the 
four  quadrics  /,  k,  k',  f  and  the  quartic  J  defined  by 

apbaaj)x  (AB)  (A^x)  (Bax). 

This  is  indeed  the  jacobian  of  the  four  quadrics,  and  represents 
the  four  planes  of  the  self-conjugate  tetrahedron  (c£  Salmon,  §  233). 

*  Of.  Turnbull,  '  Quadratics  in  n  variables'  (pp.  235-238),  Camb.  Phil.  Trans., 
Vol.  XXI,  No.  viii. 


H 


of  the  Concomitants  of  Two  Quadrics  199 

Correlatively,  S,  %,  x,  S'  are  the  four  quadrics  in  u  which  make 
the  system  of  contra  variants  together  with 

j  =  a^haUaU^  i-^B)  {Ahu)  {Ban). 

This  latter  represents  the  four  vertices  of  the  same  tetrahedron. 
In  fact,  the  jacobian  of  u^,  u^,  (^Ahuf,  (Bau)"  is 

(a/3  Ab  Ba)  Ua,u^  {Abu)  {Bau), 

where  A  =  a  a",  say ;  expanding  the  first  bracket  this  becomes 

iijd^"  (bBa)  M -  d/  b^  {ci"Ba)  M  +  &„d/  {ci"Ba)  M, 

where  each  term  represents  two,  with  a,  a"  permuted,  and  M  is 
short  for  UaU^{Abu){Bau).  But  the  factor  Ua  is  reducible  to  Ua' 
(Gordan,  il,  §  6) ;  which  means  in  this  case  that  the  symbols  u  of 
the  factors  u^,,  (Abu)  would  be  bracketed.  Hence  the  product  in- 
volving cia  is  zero.    Thus  the  jacobian  is  equal  to 

baCip'(d"Ba)M, 
=  b^a^  {a"Ba')M+ba.h'  {a"ab"a)  M    (if  B  =  b'b") 
=  —  ba a^ (a'a'B) M  (as  before) 

A  correlative  reduction  applies  to  the  case  of  /. 

The  complexes. 

§  7.  A  complex  is  a  function  of  ]),  or  line  coordinates,  but  not 
explicitly  of  ii  or  x.  There  are  eight  quadratic  and  eight  cubic 
complexes  in  the  system.    The  quadratics  are 

(ApY  or  n,  (Bjyf  or  TI',  (abpf  or  tt,.,,  (a/3p)'  or  IIi., 
{AB){Ap){Bp)  or  C,  {abp){a^p)a^b„.,  F;'  and  Fi. 

Differentiation. 

I  8.    Let  p  be  any  symbolic  product  belonging  to  the  whole 

dP 
system;  then  — -  {i=l,  2,  3,  4)  would  be  composed  of  terms  each 

OXi 

with  one  odd  S}?mbol  Qj  or  6,:  left  over.    Thus  the  four  symbols 

^:—  may  be  considered  as  the  coordinates  of  a  certain  plane.    For 

dxi       ^  ^ 

example  the  coordinates  of  the  polar  plane  of  a  point  (x)  with 
regard  to  a^-  are  («a;«i,  axU.,,  a^a^,  a^ai).  Likewise  ^ —  would  give 
a  set  of  point  coordinates. 


200  Mr  TurnhuU,  Some  Geometrical  Interpretations 

Again,  ^ —  would  give  six  quantities  which  would  symbolise 

the  coordinates  of  a  certain  linear  complex :  and,  m  some  special 
cases,  the  coordinates  of  a  straight  line.    For  example, 

is  a  useful  way  of  denoting  the  six  quantities  (Aj))  A^  {i,j  =  1,2, 3, 4), 
which  represent  a  straight  line,  since  they  satisfy  the  identical  re- 
lation existing  between  the  six  ^^-coordinates  of  a  straight  line. 

Li7ie  coordinates. 
§  9.    This  identity  satisfied  by  line  coordinates  (p)  is 

^pijPM=0  (1), 

which  we  denote  by  co  (p)  =  0.  Symbolically,  the  condition  that 
two  lines  p  and  q  should  intersect  is  {pq)  =  0.  If  p  is  the  line 
common  to  two  planes  u,  v,  and  q  is  that  common  to  u',  v',  then 
this  condition  is  (iivuv)  =  0. 

If  two  lines  p,  q  intersect,  then  Kpij  +  Xqij  represents  the  co- 
ordinates of  any  line  of  the  plane  p,  q  passing  through  the  connnon 
point  of  p,  q.  Since  the  line  p  touches  the  quadric  /  if  {Ap)-  =  0, 
it  follows  that  the  line  (k,  \)  touches  this  quadric  if 

k'  (Apf  +  2k\  (Ap)  (Aq)  +  X2  (^Aqf  =  0. 

Hence  (Ap)(Aq)  vanishes  if  p  intersects  the  conjugate  of  q  in  f; 
for  then  p  and  q  are  harmonic  conjugates  of  the  two  tangents  to  / 

r)TT 

in  this  pencil  of  lines  (k,  \).    This  shews  that  the  coordinates  ^::—  , 

op 

i.e.  (Ap)Aij,  are  those  of  the  line  conjugate  to  p  in  the  quadric  f. 

Analytically  it  is  evident  that  these  coordinates  represent  a  line 

and  not  a  linear  complex,  since  they  satisfy  the  required  condition 

(1).    In  fact 

(Ap)  (A'p)  (A  A')  =  A  {AAy  <o  (p^. 

But  the  left  member  of  this  equation  is  the  symbolic  equivalent  of 
substituting  (ApJAy  forp  in  (1):  which  proves  the  statement. 

Complexes  and  their  polars. 

I  10.  Let  (Dpy  =  0  represent  one  of  the  quadratic  complexes 
of  §  7.    Then  (Dp)  By  gives  the  coordinates  of  a  linear  complex 

*  Cf.  Gordan,  ii,  §  6. 


of  the  Concomitants  of  Two  Quadrics  201 

pola7'  to  (p)  in  (Dpf.  If  (p)  is  a  member  of  the  complex  (Dp)-, 
the  polar  is  called  the  tangential  linear  complex. 

The  complex  (Dp)Dij  is  not  usually  a  special  linear  complex. 
The  preceding  case  was  exceptional.  For  in  that  case  the  quad- 
ratic complex  was  (Apf  =  0,  and  all  the  rays  touched  the  quadric/. 

The  comptlexes  tt,.,  IIi.,,  C,  (ahp){oL^p)a^ba. 
§  11.    The  principal  quadratic  complexes  which  occur  are 
'jr,,  =  (abpy,    U,,=  ial3pY,    G^{AB){Ap){Bp). 

The  two  former  are  well  known,  ttjo  being  the  aggregate  of  lines 
cutting  the  quadrics  harmonically,  and  rTi.  being  the  correlative 
complex.  The  third,  G,  is  the  complex  of  lines  whose  conjugates, 
in  /and  /'  respectively,  intersect.  For  the  conjugate  of  -p  in  /  is 
{Ap){A)  and  in  /'  is  {Bp){B).  Again,  G  is  satisfied  too  by  the 
singular  lines  of  the  complex  ttj.,.  For  if  p  is  a  line  of  {ahp)-  =  0, 
its  tangent  linear  complex  (§  10)  is  {ahp)  (abq)  =  0 ,  q  representing 
current  coordinates :  further,  jj  is  a  singular  line  if  this  tangent 
linear  complex  is  special,  i.e.  if 

(abp){aba'b')(a'b'p)  =  0, 

which  reduces  to  (AB)  (Ap)  (Bp)  =  0.  Correlatively  C  also  contains 
the  singular  lines  of  the  complex  ITi.,. 

Again,  the  singular  lines  of  the  complex  G  belong  to  the  com- 
plex (abp)  {a^p)  Ufiba.  This  follows  in  the  same  way  as  in  the  above 
case.  But  a  more  direct  interpretation  of  this  last  form  arises  from 
the  apolar*  condition  for  two  linear  complexes  ;  if  the  polar  linear 
complexes  of  a  line  (p)  with  regard  to  ttjo  and  ITis  are  apolar,  then 
(abp) (a^p) cipba  vanishes. 

The  complexes  F{~,  Fi. 

§  12.  Besides  the  original  complexes  {Apy  and  {Bpy,  and  the 
four  complexes  of  §  11,  there  remain  two  more  quadratics,  i'V  and 
F^.  Just  as  {abpf  is  the  harmonic  complex  between  /  and  /',  so 
F^^  is  the  harmonic  complex  between  /'  and  k,  while  F^  is  that 
between  /  and  k'.  To  prove  this  we  build  up  a  form  (/',  ky  from 
/'  and  k,  in  the  same  way  as  (/  f'f,  i.e.  {ahpY,  is  built  from  /  and 
/.     Then 

{f',ky^{b,^,{ABxrf 

=  (bx',  2ft^"^aa;'-—  la^a^'axaxY 
=  2ap^  (abpf  —  2a^a^'  {abp)  {abp) 
=  [{abp)  a/  -  {a'bp)  a^J  =  F,^    (|  2). 
*  The  linear  complexes  {Dp)  —  0,  {Ep)  —  0  are  apolar  if  {DE)  -  0 . 


202  Mr  T'urnbull,  Some  Geometrical  Interpretations 

The  eight  cubic  complexes : 

F,  (ahp)  K  (Bp),  F,  (oL^p)  ap  (Bp), 

F,  (ahp)  K  (AB)  (Ap),  F,  {oc^p)  a^  (AB)  (Ap) ; 
and  four  involving  F^. 

§  13.  If  aj^,  hx,  Cx  are  three  quadrics,  the  lines  p  cutting  them 
in  involution  are  given  by  the  cubic  complex 

{hep)  (cap)  (ahp)  =  0. 

Let  us  denote  this  complex  by  the  symbol  {a^,  h^',  c^)-     Then 
{f,f',  k')  may  be  formulated,  and  we  shall  have 

(ax%  hx^,  k')  =  {{ahp)axb^,  {Baxff 

=  ((abp).axbx,  -  2b:bx"ba"bx  +  "^^'b^'J 

=  -  2  {abp)  {ab"p)  (bb'p)  bjb^'  +  2  {abp)  (ab'p)  (bb'p)  b^"\ 

The  second  term  is  zero,  since  b,  b'  are  interchangeable.     The  first 
term  is  F^  (abp)  ba.  (Bp)  to  a  constant  coefficient. 

Reciprocally  (S,  ^',x)  represents  F.2  (a^p)  a^  (Bp) ;  and  there 
are  two  like  forms  involving  Fi. 

§  14.  This  leaves  four  complexes  such  as  F2  (abp)  ba  (AB)  (Ap) 
to  be  interpreted,  but  the  geometrical  significance  is  not  at  all 
immediate.  If  however  we  write  (/,  /',  k')  as  (Bpf,  then  the  line 
(p)  has  a  polar  linear  complex 

(npf(Dq)  =  0. 

And  if  q  =  (Ap)(A),  i.e.  if  q  is  the  conjugate  line  of  p  in  the 
quadricy,  then 

(Dpy(DA)(Ap)  =  0. 

This  latter  form  is  equivalent  to  F2  (abp)  ba.  (AB)  (Ap)  :  and  similar 
results  follow  for  the  other  three  forms,  as  in  §  13. 

The  mixed  concomitants. 

§  1.5.  To  denote  the  order  of  a  form,  let  (i,j,  k)  mean  that  the 
order  is  i  in  x,  j  in  p,  and  k  in  u.  Then  there  are  three  linear 
forms  (1,  0,  1)  and  sixteen  linear  forms  (1,  1,  1). 

The  three  linear  forms  (1,  0,  1) : 

Up  %  ax,    Uahabx,    (AB)(Ab  u )  bx'. 

If  (v)  is  the  polar  plane  of  a  point  (x)  in  /,  then  (v)  =  ax(a). 
Hence  Usa&av.  =  0  is  the  condition  that  a  conjugate  plane  of  u  in/' 


of  the  Concomitants  of  Two  Quadrics  203 

should  be  the  polar  of  x  in  f  Similarly  for  Uahahx-  Again, 
{AB){Ahu)  h^  vanishes  if  the  polar  of  x  in/'  is  conjugate  to  u  in  ;)^, 
i.e.  in  {Ahiif  =  0. 

The  sixteen  forms  (1,  1,  1) : 

two  like  ax  {Ban)  (Bp),  two  like  Ua  {Bax)  (Bp), 

„    a3^a^{a^p)ua,  „      „    ax(abp)baUa, 

„      „    ax{Bau)(AB)(Ap),       „      „    u^(Bax){AB)(Ap), 
„    (abp){Abu)(A/3x)ap,    „      „    (oL^p)(Abu)(A/3x)ba. 

§  16.  The  polar  plane  of  a  point  (x),  with  regard  to  /,  meets 
a  plane  (u)  in  a  straight  line  whose  coordinates  are  (au)  a^.  If 
ax(aBu){Bp)  =  0,  this  line  cuts  the  conjugate  of  p  in  /'.  Let  us 
denote  this  relation  by  (f^,  11').  The  significance  of  the  reciprocal 
of  this,  viz.  (Stt,  n'),  is  obvious.  This  accounts  for  four  forms  since 
either /or/'  can  be  employed. 

Suppose  we  word  this  relation  differently  and  say  that  the 
plane  {u)  cuts  the  polar  of  (x)  in/ in  a  line  which  lies  in  the  linear 
complex  polar  of  (p)  in  II' :  then  a  like  meaning  attached  to 
(fx,  His)  interprets  tta;%  (^^p)  Ua..    So  also 

i-u,  'ir^^  =  ax{abp)ba.Ua, 
(/,,  {AB) {Ap) (Bp))  =  a, (aBu)  (AB)  (Ap), 

with  reducible  terms,  and 

(tu,  (AB)  (Ap)  (Bp))  =  lu  (Bav)  (AB)  (Ap), 

while  (kx,  TTia),  (Xu>  II 12)  denote  the  remaining  two  forms  of  the 
above  list.  To  complete  the  set  of  sixteen  forms  we  merely  write 
%'  for  S,  k'  for  k,  and  so  on. 

The  polar  quadrics  (0,  1,  2)  and  (2,  1,  0). 

I  17.  There  are  nine  forms  of  order  (2,  1,  0),  any  one  of  which 
represents  a  quadric  associated  with  a  given  line  (p) ;  or,  from 
another  point  of  view,  represents  a  linear  complex  associated  with 
a  given  point  (x).  The  simplest  of  these  is  (abp)  ajj^.  Let  this 
denote  the  polar  quadric  of  the  line  (p)  with  regard  to  the  system 
/*+  X/'.    It  is  convenient  to  use  the  symbol  p  (ff)  for  this  relation. 

The  equation  (abp)  aj)x  =  0  is  the  analytical  condition  required 
when  the  polar  planes  of  a  point  (x)  with  regard  to  /  and  /'  meet 
in  a  line  which  intersects  (p).  For  the  coordinates  of  these  polar 
planes  of  x  are  denoted  by  a-^a^,  bib^  (*  =  1,  2,  3,  4).  Hence  the 
coordinates  of  their  line  of  intersection  are  aj)x(ah)ij;  and  this 
line  cuts  (p)  if  (abp)  axbx  =  0. 


204  Ml'  Turnhull,  Some  Geometrical  Interpretations 


■ 


Forming  the  invariant  of  the  polar  quadric,  we  obtain  an  ex- 
pression which  reduces  directly  to  {{AB)(A2)){Bp)\^.  Hence  if  j9 
belongs  to  the  complex  G,  its  polar  quadric  is  a  cone. 

§  18.  Again,  the  tangential  equation  of  the  polar  quadric 
(abp)  cij)y;  =  0  is  formed  in  the  same  way  as  Ua~  is  formed  from  a^-. 
A  simple  reduction  leads  to 

{Ap)  (Bp)  (abp)  (aBu)  (bAii). 

Likewise  the  point  equation  of  {ajBp)  u^  a^  involves  the  form 

(Ap)  (Bp)  (a^p)  {AjSx)  {Box). 

This  interprets  the  two  forms  of  orders  (0,  3,  2)  and  (2,  3,  0). 

1 19.  Again,  if  we  form  the  polar  quadric  oi  {p)  with  regard  to 
each  pair  of  quadrics /, /',  h,  k',  we  obtain  the  following  results : 

p  (/,  k)  equivalent  to  (Ap)  (A /3a;)  a^a^,  with  a  like  form  for  p  (f,  k'), 

p{f,k')  „  F,a^(Bax),  „  „  p(f',k), 

p  (k,  k')  „  (A^x)  {cc/3p)  (Bax)  [AB). 

If,  further,  (q)  is  the  conjugate  line  of  (p)  in  {Bpy,i.e.  in/', 
then 

q  (/,  k)   is  equivalent  to  a^a^  (A/3x)  (AB)  (Bp), 
and  q'  (/',  k')  „  b^^b,  {Bax)  (AB)  (Ap). 

All    these    equivalences    are   readily  verified,  but  we   give   a 

special  proof  for  the  case  of  p  (k,  ¥).     In  fact,  the  polar  plane  of 

dk 
X  m.  k  =  0,  i.e.  in  (A^xy  =  0,  has  coordinates  — ,  which  may  be 

OXi 

symbolised  as  (A^x)(A/3)*.  So  also  the  coordinates  of  the  polar 
of  X  in  k'  are  denoted  by  (Bax)  (Ba).  Hence  the  line  of  inter- 
section of  these  polars  is  denoted  by  (Aj3x)(Bax)  [ABa^],  which 
is  equal  to  (A ^x)  (Bax)  (AB)  (a^)* ;  and  the  line  cuts  p  if 

(A^x)  (Bax)  (AB)  (aj3p)  =  0. 

§  20.  These  eight  polar  quadrics  now  enumerated,  viz.  p  (f,  f), 
p (f,  k),  ...,  q  (f,  k'),  must  be  supplemented  with  one  more  form, 
(a^p)  cipbaa^bx,  to  complete  the  set  of  nine  forms  (2, 1,  0)  belonging 
to  the  irreducible  system  of  two  quadrics  /  and  f.  The  geometrical 
significance  of  this  last  form  is  as  follows :  the  line  joining  the 
two  points,  Xi  and  x^,  cuts  p ;  Xj^  being  the  pole  in  /  of  the  plane 
whose  pole  in  /'  is  x,  and  x^  being  the  pole  in  /'  of  the  plane  whose 
pole  in  /  is  x. 

*  {A^)  =  a  a' -a'  a,  and  the  combination  of  (A^)  with  (Ba),  as  a  transvectant, 
into  [ABa^]  is  essentially  the  reduction  of  Ch.  ii,  §  15  in  the  paper  of  Gordan. 


of  the  Concomitants  of  Two  Quadrics  205 

Correlatively  there  are  nine  forms  (0,  1,  2),  quadratic  in  u, 
exactly  parallel  with  the  above,  of  which  (a^p)  UaU^  is  the  simplest. 

The  four  forms  (3,  0,  1)  and  then'  correlatives: 

(Abu)  (AjSa;)  a^axb^,  {A^x)  (Abu)  ba.Ua.u^, 

(AB) (A/3x) (BoLx)  a^a.^Ua,  (AB)  (Abu) (Bau)  baiua^, 
and  four  similar  forms  interchanging  f  and  f, 

\  21.  If  a^',  bx",  Cx"  signify  any  three  quadrics,  then  (abcu)  a^b^Cx 
vanishes  when  the  common  point  of  the  polars  of  (x)  in  the  three 
quadrics  lies  on  the  plane  (u).  Applied  to  the  quadrics  ff,  k,  k' 
taken  three  at  a  time,  this  condition  involves  the  four  forms  (3, 0, 1) 
indicated  above.  The  correlative  condition,  applied  to  each  set  of 
three  from  among  S,  S',  %,  %',  gives  rise  to  the  four  forms  (1,  0,  3). 
For  example,  if  we  select  /,/',  k  as  the  three  quadrics,  then  the 
condition  is  (Abu)  (A^x) a^axbx  =  0. 

The  polars  of  (x)  in  all  four  quadrics/,/',  A:,  k'  meet  in  a  point 
if  (x)  lies  on  any  face  of  the  self-conjugate  tetrahedron 

(A^x)  (BoLx)  a^baaxbx  =  0. 

The  remaining  forms  of  the  system. 

I  22.  None  of  the  remaining  forms  appear  to  have  any  special 
geometrical  importance  :  but  we  give  a  few  examples.  First,  as  to 
the  forms  of  order  (2,  0,  2),  we  may  exhibit  them  as  follows: 

ax  (aBu)  (Bax)  lu  and  a  similar  form, 

(Abu)(A^x)baa^iL„,ax       „  „ 

(AB)  (Abu)  (Bau)  axbx  and  a  correlative  form, 
and  [(AB)J. 

Suppose  (q)  to  denote  the  common  line  of  the  plane  (u)  and 
the  polar  of  (x)  in  /  and  (q)  to  denote  the  line  joining  (x)  to  the 
pole  of  (u)  in  /.  Then  the  condition  that  q,  q  should  satisfy 
the  harmonic  relation  (Bq)  (Bq')  =  0  becomes  on  substitution 

Ux  (aBu)  (Bax)  Ua  =  0. 

Thus  the  first  in  the  above  group  of  forms  is  interpreted.  The 
second  form  vanishes  if  two  lines  (q),  (q)  satisfy  the  harmonic 
relation  (abq)  (abq')  =  0,  where  (q)  now  denotes  the  intersection  of 
the  plane  (u)  with  the  polar  of  (x)  in  k,  while  (q^)  is  the  same  as 
before. 

Again,  the  third  form  of  the  set  vanishes  if  the  lines  in  which 


206      Mr  Turnhull,  Some  Geometrical  Interpretations,  etc. 


the  polars  of  {x)  in  f  and  /'  cut  the  plane  {u)  satisfy  the  har- 
monic relation  for  the  complex  C=(AB)(Ap)(Bp). 

Finally  the  last  form  [(ABYY,  which  is  equivalent,  except  for 
reducible  terms,  to  (Abu)bx  {Ab'u)hx  (§  2).  is  involved  in  the  con- 
dition that  the  line  common  to  (a)  and  the  polar  of  (x)  inf  shuuld 
touch  /. 

§  23.  Next  there  are  four  forms  of  order  (0,  2,  2),  such  as 
(Ap){Abu)  (abp)apUp,  (Ap)(AB){Bau){al3p)apUa,  and  four  cor- 
relatives of  order  (2,  2,  0).  All  of  these  have  obscure  geometrical 
properties,  though  they  present  no  difficulty  to  identify. 

After  this  there  are  twenty-four  forms  of  order  (1,  2,  1).  The 
simplest  of  these  is  (Abu)  b^  (Ap)  (Bp),  which  vanishes  Avhen 
u,  X,  p  satisfy  the  following  conditions :  if  the  polar  of  (x)  in  /' 
meets  (p)  at  a  point  {y),  and  if  the  polar  o{(y)  in/'  cuts  the  plane  (u) 
in  a  line  (q),  then  p,  q  satisfy  the  harmonic  relation  {Ap)  {Aq)  =  0. 
The  remainder  of  these  (1,  2,  1)  forms  are  of  like  nature. 

Beyond  this  there  are  four  forms  (2, 1,  2),  and  two  forms  (3,  0,  3), 
none  of  which  present  concise  geometrical  interpretations. 


i 


Mr  Ramanujan,  Some  properties  o/p(n) 


207 


Some  properties  of  p  (n),  the  number  of  partitions  of  n.  By 
S.  Ramanujan,  B.A.,  Trinity  College. 

[Received  3  October  1918 :  read  28  October  1918.] 

§  1.  A  recent  paper  by  Mr  Hardy  and  myself*  contains  a  table, 
calculated  by  Major  MacMahon,  of  the  values  oip{n),  the  number 
of  um-estricted  partitions  of  n,  for  all  values  of  n  from  1  to  200. 
On  studying  the  numbers  in  this  table  I  observed  a  number  of 
curious  congruence  properties,  apparently  satisfied  by  p  {n).   Thus 

(1)    |)(4),        p(9),        23(14),      p(19),    ...  =  0(mod.  5), 


(2) 

p{^\ 

P(12), 

p{n\ 

i^(26),    . 

.  =  0  (mod.  7), 

(3) 

P(6), 

P(17X 

P(28), 

i^(39),    . 

.  =  0(mod.  11), 

(4) 

i^(24), 

yo(49), 

^3(74), 

p{m,  . 

..  =0(mod.  25), 

(5) 

p{^% 

P  (54), 

^(89), 

i>(124),. 

..  HO(mod.  35), 

(«) 

i>(47), 

i9(96), 

/;(145), 

i>(194), . 

..  =  0(mod.  49), 

(7) 

p(39), 

i^(94), 

^(149), 

=  0  (mod.  55), 

(8) 

pi^n 

p(138). 

=  0  (mod.  77), 

(9) 

^(116), 

B0(mod.  121), 

(10) 

jt^(99),. 

=  0(mod.  125). 

From  these  data  I  conjectured  the  truth    of  the  following 
theorem : 

If  h=  b'^mV  and  24A,  =  1  (mod.  S),  then 

p{\),  p(\  +  8),  p{X  +  28),...  =  0(mod.B). 

This  theorem  is  supported  by  all  the  available  evidence ;  but 
I  have  not  yet  been  able  to  find  a  general  proof. 

I  have,  however,  found  quite  simple  proofs  of  the  theorems 
expressed  by  (1)  and  (2),  viz. 

(1)  p  {57n  +  4)  =  0  (mod.  5) 

and  (2)  p  (7m  +  5)  =  0  (mod.  7). 

*  G.  H.  Hardy  and  S.  Ramanujan,  'Asymptotic  formulae  iu  Combinatory 
Analysis',  Proc.  London  Math.  Soc,  ser.  2,  vol.  17,  1918,  pp.  75—115  (Table  IV, 
pp.  114—115). 


VOL.   XIX.   PART  v. 


15 


208  Mr  Ramamijan,  Some  properties  of  p  (»), 


From  these 

(5)  p  (35m  +  19)  =  0  (mod.  35) 

follows  at  once  as  a  corollary.     These  proofs  I  give  in  §  2  and  §  3. 
I  can  also  prove 

(4)  p  {Ton  +  24)  =  0  (mod.  25) 

and  (6)  p  (49n  +  47)  s  0  (mod.  49), 

but  only  in  a  more  recondite  way,  which  I  sketch  in  §  3. 

§2.     Proof  of  {\).     We  have 

(11)  X  {(1  -  cc) {l-x-){l- x^). ..Y 

=  X  (1  -  3«  +  haf  -1x^^  ...){\-x-x^^  X'  +...) 
=  t(-  lY+^(2/ji+l)x^+^-'^'i^+'>+i''^^''  +  ^\ 

the  summation  extending  from  ^  =  0  to  fi  =  x  and  from  v  =  —  oc  to 
y  =  00  .     Now  if 

l+i/A(/x+l)  +  ii^(3i^  +  l)  =  0(mod.  5) 
then  8  +  4/x  (/x  +  1)  +  4z^(3!^  +  1)  s  0  (mod.  5), 

and  therefore 

(12)  (2ya  +  l)^  +  2(i;  +  l)^  =  0(mod.  5). 

But  (2/x  +  1)^  is  congruent  to  0,  1,  or  4,  and  2(v  +  1)-  to  0,  2,  or  3. 
Hence  it  follows  from  (12)  that  '2/x  +  l  and  v  i-1  are  both  multiples 
of  5.  That  is  to  say,  the  coefficient  of  x^^^  in  (11)  is  a  multiple  of  5. 
Again,  all  the  coefficients  in  (1  —  x)~^  are  multiples  of  5,  except 
those  of  1,  x^,  x^°,  ...,  which  are  congruent  to  1 :  that  is  to  say 

(1  —  xf      1  —x" 

1  —x^ 

or  -z rr  =  1  (mod.  5). 

Thus  all  the  coefficients  in 

(1  ~  x')  (1  -  X'')  (1  -x'')... 


1 


{{l-x){l-x"~){l-x')  ...Y 
(except  the  first)  are  multiples  of  5.    Hence  the  coefficient  of  x^'^  in 
^(l_^..)(l_^ao)...       _  (l-a^)(l-x-)^.^ 

{l-x)(l-x^){l-a^)...~^^        ^^        ""'^'--^  {(l-x){l-x')...Y 
is  a  multiple  of  5.     And  hence,  finally,  the  coefficient  of  .r"'  in 


(1  -x){l-  X')  (1  -  x^) 
is  a  multiple  of  5  ;  which  proves  (1). 


the  number  of  partitions  of  n  209 

§  3.     Proof  of  (2).    The  proof  of  (2)  is  very  similar.    We  have 
(13)     a-\(l-x){l-af)il-x^..:f 

=  x-  (1  -  3a;  +  5x'  -  7a-«  +...)- 
=  S  (-  1)'^  +  "  (2//.  +  1)  (2i/  +  1)  a,-+i'^<'^+i>  +^v(.  +  i)^ 
the  summation  now  extending  from  0  to  oo  for  both  jx  and  v.     If 

2  +  l/^(^  +  l)  +  iz/(z;  +  l)  =  0(mod.  7), 

then  16  +  4/A(ya  +  l)  +  4z/(z7  +  l)  =  0(mod.  7), 

(2/A  +  l)"  +  (2i^-l-l)-  =  0(mod.  7), 

and  2/x  +  1  and  21^  +  1  are  both  divisible  by  7.    Thus  the  coefficient 
of  .^■™  in  (13)  is  divisible  by  49. 
Again,  all  the  coefficients  in 

(l-a;7)(l-a;»)  (l-a;^!)... 

[{I  - x)J\ -^x^) {V-a?)  ... Y 

(except  the  first)  are  multiples  of  7.    Hence  (arguing  as  in  §  2)  we 
see  that  the  coefficient  of  a'""  in 


(l-^-)(l-^')(l-^')--- 
is  a  multiple  of  7  ;  which  proves  (2).     As  I  have  already  pointed 
out,  (5)  is  a  corollary. 

§  4.  The  proofs  of  (4)  and  (6)  are  more  intricate,  and  in  order 
to  give  them  I  have  to  consider  a  much  more  difficult  problem, 
viz.  that  of  expressing 

p  {X)  +  jj  (\  +  h)  X  +  j)  (A,  +  2S)  X  +  . . . 

in  terms  of  Theta-functions,  in  such  a  manner  as  to  exhibit  ex- 
plicitly the  common  factors  of  the  coefficients,  if  such  common 
factors  exist.  I  shall  content  myself  with  sketching  the  method 
of  proof,  reserving  any  detailed  discussion  of  it  for  another  paper. 
It  can  be  shown  that 

(14)  ^^  "  ^'^  ^^  "  ^"^  (1  -  ^1 .  ^. 1 


{l-x''){l-x^){l-x'^)  ...       ^-'-x'  -^X' 

_  |-^  -  Zx^  +  or  (^-'  +  2x^-')  +  ^^(2g-^  -  ,rp)  +  x^{3^-'  +  xl')+5x^ 
~  ^  -'  -  1 1 X  -  x'^'  ' 

where  P  =  ^^  ^  "  ^1  ( ^  Z  ^(LtI^L  • ' 

^      {\-x^)(l-x'){l-x''){l-af)...' 

the  indices  of  the  powers  of.'?;,  in  both  numerator  and  denominator 

15—2 


210  Mr  Ramanujan,  Some  properties  of  p  (n) 


of  |,  forming  two  arithmetical  progi'essions  with  common  difl-erence 
5.     It  follows  that 

(15)     (l-oc^)  (1  -  x^°)  (1  -  X'')  ...{p{4<)+p(9)a;  +  jj(14).7;-+  . ..} 

5    . 

Again,  if  in  (14)  we  substitute  cox%  (o^x%  (o"x\  and  q}*x%  where 
w"  =  1,  for  x^,  and  multiply  the  resulting  five  equations,  we  obtain 

\(l-ay^){l-x^^)(l-x^^)...Y^  1 

^^^      \(l-x){l-x'^)(l-x')...  I       ^-'-Ux-x'^-^- 
From  (15)  and  (16)  we  deduce 
(17)    ^(4)+_p(9)^'+_p(14)a;-+ ... 

_     {(1  -  x'){l  -  x'^yjl-x^')...}'  . 


I 


{(1  -x){l  - x')(l-a^)  ...Y  ' 
from  which  it  appears  directly  that  ^ (5m  +  4)  is  divisible  by  5. 
The  corresponding  formula  involving  7  is 

(18)    p(5)+p(12)x  +  p(19)x''+  ... 

{(1-^)(1-^'^)(1-^^)...}^ 

-  ,  ^9^  {a-x')(l-x^^)(l-af^)...Y 
^  {{l-x)(l-x'){l-x'')...Y  ' 

Avhich  shows  that  p  (7m  +  5)  is  divisible  by  7. 
From  (16)  it  follows  that 
p (4) X  +  p  (9) x'- +  p (14!) x^  +  ... 

5{{l-x^){l-x^'>)(l-x'')...Y 

X  {l-x'){l-x'">){l-x'')., 


{l-x)(l-x'')(l-af)...  {(l-x){l-x^){l-x')...Y' 

As  the  coefficient  of  «^'*  on  the  right-hand  side  is  ^  multiple  of  5,  it 
follows  that  p  {25m  +  24)  is  divisible  by  25. 
Similarly 
p(5)x  +  p  (1 2)x'-+p(19)x^+  ... 

7{(1-«0(1-^")(1-^'')---}' 

(l-x')(l-x^')... 


=  x(l-3x+5x^-  7a'«  +  . . .) 


Ki-^)(i-^^)-r 


^^ja-^)(i-^^) 


{(1-^)(1-^-) 


18    ' 


"J 


from  which  it  follows  that  p  {4<9m  +  47)  is  divisible  by  49. 

[Another  proof  of  (1)  and  (2)  has  been  found  by  Mr  H.  B.  C.  Darhng,  to  whom 
my  conjecture  had  been  communicated  by  Major  MacMahon.  This  proof  will  also 
be  published  in  these  Proceedings.     I  have  since  found  proofs  of  (3),  (7),  and  (8).] 


Prof.  Rogers  &  Mr  RamanKJan,  Proof  of  certain  identities     211 


P roof  of  certain  identities  in  combinatory  analysis:  (1)  by  Prof. 
L.  J.  Rogers  ;  (2)  by  S.  Ramanujan,  B.A.,  Trinity  College.  (Com- 
municated, with  a  prefatory  note,  by  Mr  G.  H.  Hardy.) 

[Received  3  October  1918 :  read  28  October  1918.] 

[The  identities  in  question  are  those  numbered  (10)  and  (11)  in 
each  of  the  two  following  notes,  viz. 

q  q*  q^ 

+  l-l  +  (I-g)(l-5'=)'^(l-5)(l-f/)(l-9^)  +  -" 

= 1 (1) 

and 

q^  q^  512 

+  n^  +  (i  -q){i-  f)  +  j\-~-^(y-f)  (1  -  f)  +  •  •  • 

= ^ ...(2). 

(1  -  r/)  (1  -  (/)  (1  -  f)  (1  -  (/)  (1  -  q^-^)  (1  -  f/0      ^   ^ 

On  the  left-hand  side  the  indices  of  the  powers  of  q  in  the 
numerators  are  n^  and  n  (n  +  1 ),  while  in  each  of  the  products  on 
the  right  hand  side  the  indices  of  the  powers  of  q  form  two  arith- 
metical progressions  with  difference  5. 

The  formulae  were  first  discovered  by  Prof  Rogers,  and  are 
contained  in  a  paper  published  by  him  in  1894*.  In  this  paper 
they  appear  as  corollaries  of  a  series  of  general  theorems,  and, 
possibly  for  this  reason,  they  seem  to  have  escaped  notice,  in  spite 
of  their  obvious  interest  and  beauty.  They  were  rediscovered 
nearly  20  years  later  by  Mr  Ramanujan,  who  communicated  them 
to  me  in  a  letter  from  India  in  February  1913.  Mr  Ramanujan 
had  then  no  proof  of  the  formulae,  which  he  had  found  by  a  process 
of  induction.  I  communicated  them  in  turn  to  Major  MacMahon 
and  to  Prof.  O.  Perron  of  Tubingen ;  but  none  of  us  were  able  to 
suggest  a  proof;  and  they  appear,  unproved,  in  Ch.  3,  Vol.  2,  1916, 
of  Major  MacMahon's  Combinatory  Analysis'^. 

Since  1916  three  further  proofs  have  been  published,  one  by 

*  L.  J.  Rogers,  '  Second  memoir  011  the  expansion  of  certain  infinite  products', 
Proc.  London  Math.  Soc,  ser.  1,  vol.  25,  1894,  pi?.  318—343  (§  5,  pp.  328—329, 
formulae  (1)  and  (2)). 

t  Pp.  33,  35. 


212    Prof.  Rogers  d-  Mr  Ramanujan,  Proof  of  certain  identities 

Prof.  Rogers*  and  two  by  Prof.  I.  Schur  of  Strassburgf,  who  appears 
to  have  rediscovered  the  formulae  once  more. 

The  proofs  which  follow  are  very  much  simpler  than  any  pub- 
lished hitherto.  The  first  is  extracted  from  a  letter  written  by 
Prof  Rogers  to  Major  MacMahon  in  October  1917 ;  the  second 
fi-om  a  letter  written  by  Mr  Ramanujan  to  me  in  A.pril  of  this  year. 
They  are  in  principle  the  same,  though  the  details  differ :|:.  It 
seemed  to  me  most  desirable  that  the  simplest  and  most  elegant 
proofs  of  such  very  beautiful  formulae  should  be  made  public  with- 
out delay,  and  I  have  therefore  obtained  the  consent  of  the  authors 
to  their  insertion  here. 

It  should  be  observed  that  the  transformation  of  the  infinite 
products  on  the  right-hand  sides  of  (1)  and  (2)  into  quotients  of 
Theta-series,  and  the  expression  of  the  quotient  of  the  series  on  the 
left-hand  sides  as  a  continued  fraction,  exhibited  explicitly  in  Prof 
Rogers'  original  paper  and  in  Mr  Raman ujan's  present  note,  offer  no 
serious  difficulty.  All  the  difficulty  lies  in  the  expression  of  these 
series  as  products,  or  as  quotients  of  Theta-series. — G.  H.  H.] 

1.     {By  L.  J.  Rogers.) 

Suppose  that  \q\<l,  and  let  F,,^  denote  the  convergent  series 
(1  -  ^»^)  -  ^«(^«+i-'«  (1  -  x'"^q"''')  C\ 

where 

.,  _  (1  -  ^)  (1  -  xq)  (1  -  iC(f) ...  (1  -  xq'-^) 
^'-     (i_5)(i_5-.)(i_23^...(l_5.)     -■ 

the  general  term  being 

Then 

V,n  -  Fm-i  =  ^*"~'  (!-«;)-  x"q''+'-'^  {(1  -q)  +  x'>''~Hf"-'  (1  -  Ar^)}  d 

_,_  ^in^m+z-im  |(^1  _  ^^2)  ^  ^m-i^im-2  (^l  _  ^^2^J  (J^^ (•  2^_ 

Suppose  now  that  the  symbol  77  is  defined  by  the  equation 

vf{«^)  =f(xq)^ 
Then    ( 1  -  f)  C,  =  (1  -  x)  rj  C,-! ,   (1  -  xq^)  0,.  =  (1  -  x)  rj  C,. 

*  L.  J.  Kogers,  '  Ou  two  theorems  of  Combinatory  Analysis  and  some  allied 
identities',  Proc.  London  Math.  Soc,  ser.  2,  vol.  16,  1917,  pp."  315—336  (pp.  815— 
317). 

t  I.  Schnr,  '  Ein  Beitrag  zur  additiven  Zahlentheorie  uud  zur  Theorie  der 
Kettenbriiche',  Berliner  Sitzungsberichte,  1917,  No.  23,  pp.  301—321. 

J  I  have  altered  the  notation  of  Mr  Eamanujan's  letter  so  as  to  agree  with  that 
of  Prof.  Rogers. 


ill  combinatory  analysis  213 

Hence,  arranging  (2)  in  terms  of  t^C'i,  776*2,  ...,  we  obtain 
V   -V 

'  m        '  in—\ 
\  —  X 

=  a.'"^-i  {(1  -  ^.n-«i+ig»i-"i+i)  _  .^H^«+?n  (1  _  ^^n-m-H  ^jin-sm+i^  r]Ci+  ...} 

=  a;'''-'vVn-n,+i (3). 

CO 

If  we  write  v^^  TI  (1 —  *•(/'■)=  F,,^     (4), 

>-  =  0 

then  (3)  becomes      v„,  —  v,n-i  =  ^''"^~'^  vVn-m+i     (o). 

It  should  be  observed  that  Fo  and  Vo  vanish  identically. 

In  particular  take   n  =  2,m  =  l,  a,nd   n  =  2,  ni  =  2.     We  then 
obtain  Vi  =  7]Vo,    V2  —  v^  =  xrjv^ ; 

and  so  Vi  —  7]Vi  =  ocqr)'-v^  (6). 

Now  let  Vi  =  l  + a^x+a2cc-+ (7). 

Then  from  (5) 

1  +  aiX  +  a-iO?  +  . . .  —  (1  +  a^xq  +  a^xif  +  . . .) 

=  a-^  (1  +  tti^y/  +  aojf-(f  +  ...) ; 

and  so  a^=—l~^     (^2  =  p. ry^ ^      (8). 

1-q  (l-q)(l-q') 

But  when  x  =  q,  C,-  =  1 ;  and  so 

V,  =  {l-q)-q^{l-q^)  +  f^{l-q^)- (!)). 

From  (4),  (6),  (7),  and  (8)  it  follows  that 

^l-q^(l-q)(l-q'^)^ 

^  a-q)-qH^-q')  +  q''('^-q')---  ..  ^^ 

(l-5)(l-2'^)(l-f/)...  ^  '^- 


Similarly  we  have 

and,  when  x  —  q, 

and  V,  =  {1-  q')  -  q^ (1  -  g«)  +  5"  (1  -  f/«)  -  ... . 


\  ^  X  X'Q^ 

77  1-^     (1- 7)  (!-(/-) 


l-q      (1-g)  (!-(/) 


214    Prof.  Roger's  S  Mr  Ramanujan,  Proof  of  certain  identities 
Thus 

{l-q){\-q'){\-t)  ^ 


2.     {By  S.  Ramanujan.) 

Let 
G{x)  =  l 

+  t^      ^^^^^  ^'      ""i    \i-q){l-q^){l~f)...{l-qn 

If  we  write  1 -a;g2''  =  1  -  (^^ +  5''(1 -^g"), 

every  term  in  (1)  is  split  up  into  tAvo  parts.  Associating  the  second 
j)art  of  each  term  with  the  first  part  of  the  succeeding  term,  we 
obtain 

1  —xq 


G  (a;)  =  {1-  xhf)  -  x'^q^  ( 1  -  x-'q') 


l-q 


+  ^q  (1    ^q  )  (i_^)(i_^.>)     {-)■ 

G(x) 

Now  consider         II(x)  =  .,    ^      -  G  (xq)    (3). 

\—xq 

Substituting  for  the  first  term  from  (2)  and  for  the  second  term 
from  (1),  we  obtain 


x^q' 
a;*g"  (1  —  xq-) 


H  {x)  =  xq  -  ^3^-  {(1  -q)  +  xc^  (1  -  xf)] 


+ 


in  combinatory  analysis  215 

Associating,  as  before,  the  second  part  of  each  term  with  the  first 
part  of  the  succeeding  term,  we  obtain 

H  {X)  =  ,rq  (1  -  ^Y/)  jl-  ccY'  (1  -  ^^q')  1  ^ 

'M   i^     ^?)(l_^)(l_5.)(l_,/)  +  ' 
=  xg{l  —  x(f)  G  {x(f)  (4 ). 

II  now  we  write  K{x)  =  ^-^--^^^^^-  , 

we  obtain,  from  (3)  and  (4), 

andso  A-(^)=l+fl^^^      (5). 

In  particular  we  have 

1^A_      t^-     1      _{\-q)0{q)  .... 

1+  T  +  1  +  ...      K{\)  G{1)        ^  " 


or 


I      q        f      _  1  —  g  —  g*  +  g'  +  g^^  — 


(7). 


1    +    1   +    1    +    .  .  .  1   -  ^2  _  ^3  ^  ^9  +  gll 

This  equation  may  also  be  written  in  the  form 

1    <!_     f__{i^q){i_::q')iX-j')(i^)i^-_thi: 

1+1+1  +  ...      {\-f){\-q^){l-q^){l-cf){\-t'-)... 

(8)- 

If  we  write 

;,,._  GJ^) 

^  '     {\-xq){l- xq') (1  - xf)  ...' 
then  (4)  becomes       F  (x)  =  F  (xq)  +  xq F  {xq"), 
from  which  it  readily  follows  that 


216     Prof.  Rogers  d'  Mr  Ramanujan,  Proof  of  certain  identities 


In  particular  we  have 

1  -  r/  -  rf  +  (f  +  ry"  -  . . 


^l-q^{l-q){l-(f)  (i_,^)(l_f/)(l_,/) 


{l-q)il-cf){l-f) 
1 


and 

1  +  ^^  +  ,g'  +  ...  =  {l-q)G(q) 


.(10), 


1_^      (l-g)(l-5^)'  •••      (l-5)(l-^-^)(l-r/), 
1  —  q  —  q^  +  q' -\-  (f^  —  ... 

(i-^)(i-9^ya-^Yy^-" 
1 


(l_^.)(l_53)(l_(^7)(l_^8)(l_^l.),.. 


.(11)- 


Mr  Darling,  On  Mr  Ramanujan's  congruence  properties  of  p  (n)  217 


On  Mr  Ramanujan's  congruence  properties  of  p  («).    By  H.  B.  C. 
Darling.     (Communicated  by  Mr  G.  H.  Hardy.) 

[Received  3  October  1918:  read  28  October  1918.] 

1.     Proof  that  p  {5ni  +  4)  =  0  (mod  5). 

Let  u  ^(l-x){l-  aJ")  (1  -x')...; 

then  by  Jacobi's  expansion 

a^' =  "ST  (- 1)"  (2»  +  1)  **"^''+'^ 


n  =  » 


so  that  in  d'-ur,  where  d  denotes  differentiation  with  respect  to  x, 
the  coefficients  are  of  the  form 

i  {n  -  1)  n  (n  +  1)  (n  +  2)  {2  (n  +  3)  -  5], 

and  therefore 

d^u^=  0  (mod  5) (1). 

Again,  in  d^u"  the  coefficients  are  of  the  form 

A^{n'  +  n-4<){n-2)(n-l)n(n  +  l)(n  +  2)\2(u  +  4')-7], 

and  therefore 

an<3=  0  (mod  7) (2). 

/1\  1  2 

Now  8-  (  -    = /ci-u  +  -  (duf  ; 

\Uj  U'  u 

also  du^  =  Zu'du,  and  d-ii^  =  ^u'^dhi  +  Qu  (du)-.     Hence 

a^(-)  =  -o\9"'''  +  (r7(9'*')' (3)5 

\uj  ?>u^  9«' 

and  thus,  by  (1),  we  have 

^'  0  ^  -  II  ^^'''^^  ^"  ^  ~  27I"  ^^'''^'  ^"^^^  ^^ ' 

so  that  8^  f-]  =  0  (mod  5)   (4). 

Again  if  Iju  be  expanded  in  powers  of  x,  and  the  operator  3* 
be  applied  to  the  resulting  sei'ies,  it  is  evident  that  the  coefficients 
of  all  powers  of  x  of  the  forms  om,  5m  +  1,  om  +  2  and  5m  +  3  will 
be  multiplied  by  a  factor  divisible  by  5 ;  but  that  the  coefficients 
of  the  powers  of  x  of  the  form  5m  +  4  will  be  multiplied  by  a  factor 
which  is  not  divisible  by  5.    Hence  it  follows  at  once  from  (4)  that 

p  {oin  +  i)  =  0  (mod  5). 


2181 3Ir  Darling,  On  Mr  Ramanujan's  congruence  properties  of  p  {n] 

2.     Proof  that  p  {7m  +  5 )  =  0  (mod  7). 
Differentiating  (3),  we  have 

d'  (-]  =  -  ^,  dHi'  +  ^^  dud-u'  +  A  9  (9"')'  ('iiocl  7 ) 

OU  SU'^  i)'U' 


=  -  ~dhi^  +  j^^didu^y  (mod  7). 
Similarl}^  having  regard  to  (2), 

a*  (-]  =  ^-  du^d^u^  +  ^a-  (du'T-  (mod  7), 

d'  (^)  =  ^.  d'vPd'w  +  ~  d'  (dtt'f  (mod  7)  (5), 

^0-^,(^-'y^l,^'('"'y('--^'^ ^6). 

Again    a'(-^)  =  -3'©H-6a.Q  =  .-.3{..8»Q}; 
SO  that,  by  (5)  and  (6), 

36  (^]  =  !_'  [43  (x'^d'u'd-u')  +  63  {a^«3«  (du'f]]   (7). 

Now  d{dit'')-  =  2dHo'dit', 

3^  (8m3)2  =  23^  i(33«.3  +  2  (d-u'^y. 
Thus,  by  (2), 

3^  (du^Y  =  Qd'u'^d'-u^  (mod  7) ; 

and  therefore,  by  (7),  we  see  that 


that  is,  by  (2), 

^x 


38  (- j  =  3  {w'd'ii'dHi']  (mod  7)  ; 


=  ai'dhi^dHi^  +  6w^d^  u^'d' u'  (mod  7) 
uj 

=  d'u^d{£c'd-a')  (mod  7)    (8). 

But  the  coefficients  in  3  (x'^a-u^)  are  of  the  form 

i  (n  -  1)  9i  {n  +  1)  {n  +  2)  [2  (n  -  3)  +  7}  {(//  -  2)  (n  +  3)  +  14}, 
and  are  therefore  divisible  by  7 ;  and  therefore,  by  (8), 
3«  (-)  =  0  (mod  7). 

Hence,  by  considerations  similar  to  those  in  the  latter  part  of  §  1, 
we  see  that 

p{7m  +  5)  =  0  (mod  7). 


Miss  Wrinch,  On  the  eicponentiation  of  well-ordered  series    219 


On  tlie  exponentiation  of  luell-ordered  series.  By  Miss  Dorothy 
Wrinch.     (Communicated  by  Mr  G.  H.  Hardy.) 

[Read  29  October  1918.] 

The  problem  before  us  in  this  paper  is  the  investigation  of  the 
necessary  and  sufficient  conditions  that  P'^  should  be  Dedekindian 
or  semi-Dedekindian  when  P  and  Q  are  well  ordered  series. 

The  field  of  P'^  is  the  class  of  Cantor's  Belegungen  and  consists 
of  those  relations  which  cover  all  the  members  of  the  field  of  Q 
with  members  of  the  field  of  P :  several  members  of  the  field  of  Q 
may  be  covered  with  the  same  member  of  the  field  of  P,  but  every 
member  of  the  field  of  Q  is  covered  with  one  member  of  the  field 
of  P  and  one  only.  In  order  to  prove  that  P'^  is  Dedekindian  it  is 
necessary  to  prove  that  every  sub-class  of  the  field  of  P^  has  a  lower 
limit  or  minimum  with  respect  to  P^.  If  there  is  a  last  term  of 
the  series  P'^  it  is  the  lower  limit  of  the  null  class.  Unit  sub-classes 
have  their  unique  members  as  minima.  It  remains,  then,  to  con- 
sider sub-classes  with  two  or  more  members. 

Now  the  relation  P*'*  orders  two  relations  R  and  *S'  by  putting  R 
before  S,  if  R  covers  the  first  Q-term,  which  is  not  covered  with  the 
same  P-term  by  both  R  and  h,  with  a  P-term  occurring  earlier  in 
the  P-series  than  the  term  with  which  8  covers  it.  Suppose  A,  is  a 
sub-class  of  the  field  of  P'^  with  at  least  two  members.  We  will  call 
Qm'^  the  first  Q-term  which  is  not  covered  with  the  same  P-term 

by  all  \'s;  and  Tp^\  that  subset  of  X  which  consists  of  those  members 
of  \  which  cover  Q„/A,  with  that  term,  in  the  class  of  P-terms  with 
which  various  X's  cover  Qm'^;  which  occurs  earliest  in  the  P-order. 

Tp'\  will  therefore  be  contained  in  \  and  not  identical  with  it.    It 

will  be  seen  that  P'^-terms  belonging  to  Tp'\  come  earlier  in  the 
P'^-order  than  terms  of  \  not  belonging  to  it.     Constructing 

Tp'T'X 

we  get  a  smaller  subset  of  \ :  members  of  this  subset  occur  earlier 
in  P''*  than  other  members  of  X.     Continuing  this  process  with 

X,   T/X,   T/Tp'X,    Tp'Tp'iyx,...    /*,...    V,..., 

we  obtain  smaller  and  smaller  sub-classes  of  X:  if  /a  precedes  v  in 
this  order,  members  of  v  occur  earlier  in  the  P'^-order  than  members 
of  fM  which  are  not  members  of  v.      We  take  the  common  part  of 


220    Miss  Wrinch,  On  the  exponentiation  of  well-ordered  series 

all  these  subsets  of  \,  i.e.  the  class  of  relations  which  belong  to  all 
the  sets 

\,    2  P  \,    1 P  1  P  \  . . .  ', 

and  get  a  subset  of  X 

p'iT'pW^ 
which,  again,  consists  of  members  of  \  which  come  earlier  in  the 
P'^-order  than  members  of  A.  not  belonging  to  it.     Repeating  the 
original  procedure  we  get 

Tpy(%h'\   Tp'TpYiT'p)^'\, ..., 

and  so  obtain  a  series  of  sub-classes  of  A,  ordered  by  the  serial 
relation 

A  {Tp,  \), 

where  A  is  the  relation  between  /j,  and  v  when  v  is  contained  in  /i 
but  not  identical  with  it.  And  this  is  a  well-ordered  relation : 
CQjisequently  it  will  have  an  end,  viz. 

p%Tp^Ay\. 

If  this  is  not  null,  it  consists  of  a  single  member,  which  will  be 
the  minimum  of  X  in  P^.     But  if  it  is  null  we  will  put 

PQ'X  =  s'N {a^  .  ^,e  (Tp^Ayx .  iV=  (i^/.)  r  eQm VI- 

Then  PQ'X  is  a  relation  covering  a  certain  section  of  the  Q-terms 
with  P-terms  :  PQ'\  agrees  in  the  way  it  covers  the  Q-spaces  with 
each  member  yu,  of  ^  {Tp,  \)  as  far  as  QniV-  PQ''^  will  therefore 
cover  Q-spaces  up  to  z,  if  there  is  a  yu-  which  is  a  member  of  the 
field  of 

A  (Tp,  \) 

such  that  z  precedes  QmV  in  the  Q-order.  If  no  member  of  the 
field  of  J.  (Tp,  X)  agrees  in  the  covering  of  Q-spaces  beyond  a  cer- 
tain member  z  of  the  field  of  Q,  PQ'X  covers  no  spaces  beyond  z 
with  P-terms  and  for  this  reason  is  not  a  member  of  the  field  of  P'^. 
If  P  is  a  P'3-term  which  agrees  with  PQ'X  in  the  covering  of 
Q-spaces  as  far  as  it  goes,  R  precedes  all  the  members  of  X,  in  the 
P^-order ;  further,  any  member  of  the  field  of  P^,  following  R  and 
all  relations  agreeing  with  PQ'X  as  far  as  it  goes,  follows  at  least 
one  member  of  X.  Hence,  if  there  were  a  maximum  in  the  P'^- 
order  in  the  class  p  of  members  of  the  field  of  P*?  which  agree  with 
PQ'X  as  far  as  it  goes,  this  relation  would  precede  all  X's  and  any 
relation  following  it  would  follow  at  least  one  member  of  X.  If  the 
class  consists  of  one  term  R,  it  will  have  a  maximum,  namely  R 
itself:  R  will  then  be  equal  to  PQ'X  and  PQ'X  will,  therefore,  be 
the  lower  limit  of  X.     But  p  is  a  unit  class  only  when  PQ'X  covers 


Miss  Wrinck,  On  the  exponentiation  of  ivell-ordered  series    221 

the  ivhole  of  the  Q-terins  with  P-terras.  When  PQ'X  does  nut 
cover  the  whole  of  the  Q-tenns,  but  covers  Q-terms  only  up  to 
z  (say),  all  p's  will  agree  in  their  covering  of  Q-spaces  up  to  z,  and 
the  remaining  Q-spaces  will  be  covered  differently  by  different 
members  of  p.  To  get  a  maximum  of  the  p's  with  respect  to  P*^, 
we  want  a  relation  S  which  is  a  p  such  that  no  member  of  p  comes 
later  in  the  P^-order.  Now.  if  P  has  no  last  term,  every  P-term 
is  followed  by  other  P-terms.  However  S  covers  z  and  the  Q-spaces 
after  z,  by  replacing  the  term  covering  any  member  of  the  field  of 
Q  after  ^  by  a  member  of  the  field  of  P  following  it  in  the  P-order, 
we  obtain  a  relation  T  which  is  a  p  and  follows  ;Si  in  the  P'^-order. 
;S'  is,  consequently,  not  the  maximum  of  p  in  the  P^-order.  Now 
if  z  in  the  field  of  Q  is  covered  by  PQ'X,  the  term  innnediately 
following  z  will  also  be  covered  by  PQ'X.  Therefore,  if  Q  is  a  finite 
series  or  an  co,  PQ'X  will  always  cover  the  whole  of  the  Q-terms ; 
since,  as  X  has  at  least  two  members,  it  will  always  cover  one  Q- 
term.  Any  X  will  then  have  a  lower  limit  or  minimum  with 
respect  to  P*^.  In  such  cases,  P*?  will  certainly  be  Dedekindian 
with  the  addition  of  a  last  term,  whether  P  has  a  last  term  itself 
or  not. 

But  if  Nr'Q  is  greater  than  o),  it  is  possible  to  find  a  subclass 
X  of  the  field  of  P^  which  is  such  that  PQ'X  does  not  cover  the 
whole  of  the  field  of  Q. 

For,  let  1  and  2  represent  the  first  and  second  terms  in  the  P- 
series  and  let  (e.g.) 

i...hi(r)2...h2(aiii 

represent  a  relation  which  covers  the  first  ^  Q-terms  with  1,  sub- 
sequent terms  up  to  (but  not  including)  the  ^th  term  with  2,  and 
all  remaining  terms  with  1.  Such  a  relation  is  clearly  a  member 
of  the  field  of  P'^.  Consider  the  class  of  I'elations  X  which  cover 
all  Q-spaces  up  to  z  with  1,  and  all  the  Q-spaces  following  z  with 
2,  as  z  is  varied  from  the  second  Q-term  to  the  ^th,  where  ^  is  an 
ordinal  number  with  no  immediate  predecessor.  We  will  arrange 
this  class  of  relations  in  the  P^-order. 


1... 1-1(^)2. ..h2(0,22....     {^<0 

11112 h2(f),  22 

11122 f-2(0>  22 

11222 ^-2(0>  22 

12222 ^-2(0,  22 

This  class  has  no  minimum  in  the  P'^-'-order,  and  PQ'X  covers  all 


222     Miss  Wrinch,  On  the  exponentiation  of  well-ordered  series 

the  Q-places  up  to  the  ^th  with  1  and  does  not  cover  the  subse- 
quent Q-places  at  all.  It  is  therefore  not  a  member  of  the  field 
of  P^.  But,  as  we  have  seen,  every  relation  which  agrees  with 
PQ^X  as  far  as  it  goes,  and  covers  the  other  Q-places  with  any  P- 
terms  whatever,  precedes  all  X's :  and  any  member  of  the  field  of 
P^  following  this  relation,  and  all  relations  agreeing  with  PQ'X  as 
far  as  it  goes,  follows  at  least  one  member  of  X.  Thus,  e.g.,  the 
relation 

ll...hl(^),  2111 

precedes  all  X's,  and  any  relation  following  it  and  all  relations 
agreeing  Avith  PQ'X  as  far  as  it  goes  (as  e.g.  the  relation 

11211...  h  1(0,  21211...) 

follows  at  least  one  relation  belonging  to  X,  e.g.  the  relation 

11122...  \-2{0,  222... 

Thus  \  will  have  a  lower  limit  if  and  only  if  there  is  a  maximum 
among  the  relations  covering  all  places  up  to  the  ^th  with  1. 
And  this  is  the  case  when  and  only  when  P  has  a  last  term  u  (say). 
For  then  the  relation 

111 \-l{^)uiiu... 

will  be  the  lower  limit  of  X.  Thus  if  Nr'Q  is  greater  than  o),  it 
will  be  the  case  that  all  existent  sub-classes  of  the  field  of  P'^  will 
have  a  lower  limit  or  minimum  when  and  only  when  P  has  a  last 
term.  A  non-existent  subclass  (i.e.  a  subclass  with  no  members) 
will  have  a  lower  limit  or  minimum  when  and  only  when  P  has  a 
last  term.  If  Nr'Q  is  greater  than  co,  P^  is  Dedekindian  when  P 
has  a  last  term,  and  if  P  has  no  last  term  P^  even  with  the  addition 
of  a  last  term  is  not  Dedekindian.  We  thus  arrive  at  the  following 
conclusions.  When  P  and  Q  are  well-ordered  series,  (1)  P^  is 
Dedekindian  when  and  only  when  P  has  a  last  term ;  (2)  if  Nr'Q 
is  greater  than  co,  P*^  with  the  addition  of  a  last  term  is  Dede- 
kindian if  and  only  if  P  has  a  last  term ;  (3)  if  P^  is  made  Dede- 
kindian by  the  addition  of  a  last  term  when  and  only  when  P  has 
a  last  term,  Nr'Q  is  greater  than  co. 

These  propositions  will  now  be  established. 

[The  symbols  used  are  those  o/Principia  Mathematica.  Among 
the  propositions  referred  to,  those  whose  nwnhers  are  greater  than 
1  are  proved  in  P.M.,  ivhile  the  others  are  established  in  the  course 
of  this  paper.'] 

*01.     QjX  =  mmQ'y{s'X'y^eQKjl)  Df 

*-02.     Tp'X  =  XnM  {M'QJX  =  mmp's'X'Q^,'X)  Df 


Miss  Wrinch,  On  the  exponentiation  of  well-ordered  series    223 
*-03.    A=\fl{^lQ\.^Ji^\)  Df 

*1.       I- :  P,  Q  e  n  .  X  6  0  .  D  .  5'Cnv'P«  =  mm'(7^«)'X    [*207-l7] 

*11.     V:F,QeQ..\el.\C  C'F'^ .  D  T'X  =  min  (P^^)')^ 

Dem. 

h.*l7619.  'D\-:R6a'P^.Dj,.^{RPm)  (1) 

l-.(l).*205-18.     Dl-.Prop 

*'201.     \-:P,Qen.\C  C'P'i .  E  !  T/X  .  D  .  f{Tp^Ay\ 

=  B'Cnv'A  {Tp,  X)  .  A  (Tp,  X)  e  O 
Devi. 

[•02]  h.  TpeRl' A  nCh-^1  (1) 

h  .  (1)  .  *258-231  .     D  h  .  Prop 

*-202.     h  :  E  !  Tp'X  .  D  .p'(Tp^Ayx  ^eB'Tp  [*-201] 

*-203.     \-:P,Qen.XC  C'P'^  .X^eOwl.D.E!  Tp'X 
Dem. 
\-:.XQ  G'P'i  .R,S€X.X€  a'R  .  D^^  .  R'w  =  S'x:D  .XeO  vjI  :. 
[Transp]         D  I- :.  Hp  .  D  :  P,  >Sfe  A, .  D  .  g^  .  i2^«  =|=  iS^'a- .  iceQ^E  :. 

[*250'121]      D  h  :.  Hp  .  D  :  E  !  min,/^  (i-'A,'^  ~  e  0  u  1) :. 

[rOr02]         D  I-  :.  Hp  .  D  :  E  !  Q,„'X  .  E  !  Tp'X 

*-2031.     (- :  E  !  T/X  .  D  .  E  !  Q,/X  [*-02] 

*-204.       h  :  P,  Q  e  O  .  X  C  C"P'^  X  ~  e  D' Tp .  D  .  X  e  0  w  1 

[*-2()3 .  Transp] 
*-205.       f- .  Hp  r203  .  D  .  p'iTp^Ayx  e  0  u  1      [**-202-203-204] 

r211.      h  .  E  !  Q,„*\  .  D  .  s'X'QjX  ~  e  0  w  1  [*-01] 

*-212.      h  :.  E  !  Q,,,'X  :  D  :  {s'X)[Q'Q,,,'X  el -^C\s:  ReX  . 

D.P6l->Cls 
Pewi. 

[*176-19]  hz.ReC'P'i  .D'.zeC'Q.D.R'z^eO     (1) 

l-.(l).r02.     Dh.Prop 

VOL.  XIX.  PART  V.  16 


224    Miss  Wrinch,  On  the  exponentiation  of  well-ordered  series 

r213.      \-:S€l->C\s.R(iS.D.S[a'R  =  R 

Dem. 

V  .^u.v  .  uSv  .  V  €  Q.'R  .  ~  (uRv) .  D  .  g«,  v,  u' .  a  4=  w  • 
uSv  .  u'Rv  .  <^  (uRv)  .  V  €  d'R  : 
D\-:.R(lS.D:'3^u,v.  uSv  .  v  e  a'R  .  --  (uRv)  . 

D  .  g/t,  V,  n' .  u'  =j=  u  .  nSv  .  u'Sv  : 
0\-:.RQ.S.Sel->Ch.D:-^  {gu,  v  .  iiSv  .  v  e  Q'R  . 

~  (itRv)] 
D\-:.R(lS.Sel-^Ch.Di  nSv  .  v  e  a'R  .  D„,, .  uRv  : 
D\-:.R(lS.S€l-^Ch.:>:S\-a'R  =  R 

*-2131.     ^:.'3^l^.aCa's'uT.{s''!^)\-ael^C\s:0:Re^.D.R[a 

=  {s''ST)[a  =  (p''S7)[a 
Dem. 

h  .  *40-13  .  *41-44  .   Dhz.Rezy.D:  xRy  .D.x  (sV)  xj  : 

a'RQa\s'^)'.. 
'^ViaXia'R.Ret;T.':>.R\aQ.{s'zj)\a   (1) 

I-  *-213  .  (1) .  D  F  :  a  C  iVR  .Re^.  {s'^)  pet  e  1  -^  Cls  . 

:^.R\a  =  {s'7JT)\a     (2) 
h:.g;!tn-.i^ero-.Djj.  xRy  .yea: 

D  :  g[*S' .  S  e-sT  .  xSy  .yea     (3) 

I- .  (3)  .  D  h  :  a  !  OT  .  D  .  {p'^)  \a  G  (s'tsr)  \a  (4) 

f- .  (4) .  *-213  .  D  h  :.  a: !  w  .  D  :  (5^-sr)  [^a  e  1  -^  Cls  .  D  . 

(i'OT)Pa  =  (^^i3-)['a     (5) 
l-.(2).(5).  Dh.Prop 

r214.       V  :.  Hpr20.3  :  D  :  E  e  \ .  D  .  R[Q'QJX  =  {s'\)[Q'Q^,'X 

=  {p'X)  [Q'QJX  [*40-13  .  *41-44  .  r2131] 

r215.       h  :.  Hpr203  :D:Re\-  T/X  .  S  e  Tp'X  .  D  . 

R  [Q'Q^'X  =  S [Q'Q^'X .  {S'QJX) P (R'Q^'X) 
Dem. 

[r02]  \-:SeTp'X.D.SeX  (1) 

[**-01-02]  h  :  ^ !  Q^,'X  .  S  e  T/X  .ReX-  Tp'X  . 

D  .  S'Q,,,'X  =  minp  's'X'QjX  .  R'Q^'X  +  min's'X'Q^'X    (2) 
K  (1) .  (2) .  r2l4  .  D  h  .  Prop 


Miss  Wrinch,  On  the  expouentiatluit  of  luell-ordered  series    225 

*-216.     f- : .  Hp  *-208  :D:Re\-  Tp'\  .SeTi^'X.D.  SF'^  R  [*-2 15] 

*-217.     h : .  Hp  *-203  :  D  :  /x  e  ( Tp^A  y\  .  R  eX- /m  .  S  e  fi,  .D  .  SP'^'R 

Bern. 

h  .  *40-23  .     D  h  :.  p  C  (Tp^Ayx  .'■^l  p  :  fie  p  .  S  e /j,  .  ReX- fi . 

D  .  SP'^R  :D:Sep'p.R€X-p'p.  D^„, .  SF'^R   (1) 
h  .  (1) .  -r216.*258-241  .  D  I- .  Prop 

r218.     h:  }ii)*-203. '3^1  p'{Tp*Ayx.D.7Y(Tp^Ayx  =  mill {P'^yx 
Dem. 

V  .  *r217-201  .      D  I-  :  Hp  .  D  :  ^'  e  \  -p^Tp^Ayx  . 

Rep'iTp^Ayx.D.RP^S     (1) 
h.(l).  Dh.Prop 

*-31.       hi.SeX-.D:  k=p'{{Tp^Ayx  n^/Sf}  .  D  . 

ke(Tp*Ayx.S'^eTp'k 
Dem. 

[*22-43]  h  :.  SeX.  D  :  h  =  p'{{Tp^AyXn  e'S]  .  D  . 

'3_p.pC  (Tp^Ayx  .  g!  p  .  p  =  [{Tp^Ayx  n  e'S]  .  k=p'p 

[*257-125]  D  h  :.  A'^eX  .  D  :  k=p'{{Tp^Ayx  n  eSS'J  .  D  . 

a/3 .  It/p  e  {Tp^Ayx  .  p  =  {{Tp^Ayx  n  e'S]  .  k  =p'p 

[*258-211]  D  h  :.  6'  e  A  .  D  .  k  =  p'{{Tp*Ayx  n  e'S]  .  D  . 

^^e(Tp*^)'X  (1) 

l-.(l).*2o7-125. 

Dl-:.S€X.D.k  =  p'{{Tp*Ayx  n  e'^'l  .  D  . 

?p'^^e(^p*^)'x  (2) 

[*40-12]       D\-:.SeX.D.k^p'{{Tp^AyXne'S].D. 
^  e  (T^p*^ yx  n  e'S  .D^.kCr^: 

[Transp]       D\-:.SeX.D.k  =p'{(Tp^Ayx  n  e'S]  .  D  . 

16—2 


226    Miss  Wrinch,  On  the  exponentiation  of  vjell-ordered  series 

l-.*22-43.*-04.(2). 

D\-:.SeX.D.k  =|/{(rp*^)'X  n  e'S\  :  D  . 

~  (k  C  Tp'k) .  Tp'k  e  {Tp^A y\  ( 4) 

l-.(3).(4).    D  h  :.  >S'eX  .  D  .  k  =p'{{Tp^Ay\t^  e'S}  .  D  . 

r^iSeTp'k)  (5) 

|-.(1).(5).  Dh.Prop 

r32.       h  :  Hp  r203  .  /.  ?p  A, .  E  !  ?p V  ■  ::>  ■  (Qn/^)  Q  (<^m» 

Dem. 
[*-212]  l-:Hpr203.i?eX.D.^Q(Q^/X)D.i^*5el    (1) 

[**-02-203]        \-  .fiTpX.D.fjiCX  (2) 

h  .  (1) .  (2) .  O  1- :.  Hp  -r203  .f^TpX.Sefx. 

:>:zQ(QjX).D.S'zel     (3) 

[*-02]  f- :  Hp *-203  .  fjuTpX.  S e /j,  . 

D  .  S'Q,,,'X  =  minp'P  [s'X'y  -  e  0  u  1 }     (4) 

1- .  (3) .  (4)  .        I-  ::  Hpr203  .f^TpX.Se/x. 

D  :.  a/  :  (QJX)  Qz' :  uQz' .  D,„  .  ^S^(^  e  1     (5) 

K  (5) .  h  :  Hp*-203  .  /.  ?p  X  .  El  ?pV  •  ^  •  (Qn/^)  Q  (Qm'/-) 

r321.    h  :  Hp  r203  .  ^a  (.4  {Tp,  X))v.El  Tp'v  .  D  .  (Q,„V)  Q  ('Q^'i.) 

Dem. 
[**-02-203]  \-ipC  (Tp^Ayx  .'Rlp.'K^.p'p. 

D.lyl'Q.^Yp'^elyjO     (1) 
[*40-12]  h'.Xep.D.'p'pCX  (2) 

K(l).(2).  D\-:.pC{Tp^Ayx.'3_lp.'3_lp'p: 

D-.Xep.O.  i'X'Q,,yp'p  ~  e  1  u  0     (3) 

h  .  (3) .  -r02  .           D\-:.pC  {Tp^Ayx  .'3,1  p  .'Rlp'p  . 
*•  > 

D  :  Tp%  Xep.D.  s'Tp'X'Q,,yx  e  1  .  i'Tp'X'Q,^'p'pr^e  1  yj  0  : 

D\- :.  pC{Tp^Ayx.'3l  p  .^Ip'p: 

:>:Tp%Xep.D.Q,^'X^Qjp'p     (4) 


Miss  Wrinch,  On  the  exponentiation  of  well-ordered  series    227 
I- .  (1) .  rOl  .  (4) .   D  f-  :  pC{Tp^A)'\  .  g !  p  .  g !  jo'p  . 

(InP'p  ^V  !.v'X',y-eO  u  i;  .  Q,/\  =  miiV.??~e()  w  1     (5) 
I- .  (o)  ■  D  h  :  p  C  (rp*^)'\  .  a !  /9  .  a !  jt)'p  . 

r,/X,\6p.D.(Q„A)Q(Q,nyp)     (H) 
V  .  (6) .  *-82  .  *-258-24l  .  D  h  .  Prop 

*-33.       V  :  Hp  *-203  .  D  .  PQ'X  e  1  -^  Cls 

[*-04]  I- :  Hp  .  t!7  =  ^  [gyti  .  yLt  6  (r/,*4)'\  . 

|-:.Hp(l).D:i¥,iV6t^.Di,,v- 
[r201.*250-113]     D  h  :.  Hp  (1)  .  D  :  M, Ne^.  D^^,  v  ■  3/^,  ^^  - 

IJi  [A  {Tp,\)]v.v.v  \A  (Tp,  \)]  /Lt : 
[**-321-214]  D  F-  ::  Hp  (1)  :.  D  :.  i¥,  iYe  ^  .  D,^,  y  ■ 

D\-:.Rp(i):D.M,N€^.DAVMCa'N. 
M  =  N \  QM/ .  V  .  a'N  C  a'i¥ .  N  =  M\-a'N':. 
D:.Hp(l):D.if,.V6t^. 

D  .  2/  e  a'lM  n  a'N .  D  .  3f'y  =  N'y     (2) 
f-.(2).  D.Prop 

*-34.       I-  :  /x  e  ( '/p*^ )*\  .  E  !  ?p'X  .  Hp  r203  . 

[*-04]     1- .  Hp  *-203  .  E  !  Tp'tx  .fx.€(  Tp^*A  )'X  . 

D.(/>V)rW„>CPQ'X     (1) 


1 


228    Miss  Wrinch,  On  the  exponentiation  of  luell-oj'dei^ed  series 

[*-33]     h  .  Hp  .  D  .  PQ'X  €  1  -^  Cls 

|-.(l).(2).*-213.Dh.Prop 

r341.     h  :  /x  €(Tp*Ayx  .  Hp  *-208  .  p'(Tp*Ayx  . 

Dem. 
[*-201]  !-:Hp.D.'B^Cnv'^(Tp,\)  =  A: 

D  h  :  Hp  .  D  .  E  !  ?p  V  (1) 

|-.(l).r84.DK.Prop 

*-35.       f- :.  i^  [^  a^PQ'X  =  PQ'X  .  Hp *-203  .  p'(Tp^AYX  =  A  . 

ReC'P'^::)f>:SeX.Ds-RP'^S 
Dem. 

[**-84-31]         h  .  Hp  .  <Sf  e  X  .  /<:  =  p^K^P*-^ y^  '^'^\  ■ 

D^.Er'Q'Q^'x=5:r^Qm'x  (1) 

[**-34-02]         h  .  Hp  .  ^  e \  .  k=p'\{Tp^Ayx  nV'S]  . 

D^  .  gT .  T  e  fp'k .  K^e^'X  =  T'Q^'X  . 

(2''Q„^X)P(,Sf,Q,^X)     (2) 

h  .  (1)  .  (2) .  D  I-  :.  Hp  .DpiSeX.  D,,  ■  PP^>S^ 

*-4.         h  : :  Hp *-203  . p'(Tp^A yx  =  A.ze  a'PQ'X  :.D:.S ['Q'z 

=  (PQ'X)  [~Q'z  :  (PQ'X'z)  P  {S'z)  .D.'3^U.UeX.  UP'^S 
Dem. 

[r04]  h-.ze  a'PQ'X  .:>.'^pi.fMe  {Tp^AyX  .  2  e'q'Q^'iM  : 

[r341]         D  h  : .  Hp  .  D  :  ayu  .  ^  =  Tp V  ■  3  ■  (PQ'^)  [~Q'Qm'i^ 

[r341]         D  h  :  Hp  .  D  .  a/.,  f^ .  C/^  6  ?p ^  ■  (i'Q*^)  T  Q'Qm'Tp^M' 

=  U['Q'Q^'Tp'v.zQ^iQ^'v): 

[*176-19]     D  I- :  Hp  .  D  :  ,S7 Q'^  =  PQ'X [Q'z  . 

(PQ'X'z)  P  (S'z)  .  D  .  a  ^  .  [/  e  X  .  UP'iS 


Miss  Wrinch,  On  the  exponentiation  of  well-ordered  sei'ies    229 

*-41.       t- : .  Hp  *-203  .  p\Tp^-A y\  =  A  .  i2  e  C'P'? . 

7?  [  (VPQ'X  =  PQ'\.D.  RP'i  V :  D  :  g^S' .  ,S'  e  X  .  .S'pv  V 
Dem. 
[*17G-19]         h:Hp.D.-(rra'PQ'\  =  PQ'A,): 

[*176-19]     D  H  :  Hp  .  D  .  32  , 5  6  (J'PQ'X .  V  [Q'z 

=  (PQ'X)  ['Q'z  .  (PQ'X'z)  P  ( V'z) 
[*-4]  h  .  Hp  .  D  .  a^S' .  <S6\  .  SP^V 

*-42.        |-:.P,  QeO.X-eOul.xC  C'P'^' .  p'(Tp^A  Y\  : 

D:p  =  C'P'i  nR[R\ a'PQ'X  =  PQ'X]  .D.pC p'P'"'X  . 

pcpQ"p  c  s'P''\     [*r3.r-H] 

r43.        \-:¥AQ^'X.D.s'X'Q^'X^el.s'fp'X'Q^'Xel     [*r0r02] 
r431.     \-:Tp(lA.{Tp\^(lA 

Deni. 
|-.r43.  Dh:  fiTpX.D.ij,CX.^^X  (1) 

|-.(1).*201-18.  Dh.Prop 

*-432.     h  :  Hp *203  .  D  . "I^i (P«)'X  C /j'CTp*^ )'X 

[*-21 7]  h  :.  Hp  .  P  e  /ci  .  yLt  e  (^V*.! )'X  .  /x  {xl  (T'p,  \)|  v  . 

P  ~  6  z/ :  D  :  g!  z^ .  D  .  a*S' .  >SP^'P  :. 
DI-:.Hp.Peyit.yL6e  (rp*J.)'\  .  P  mill  (P'O  X  : 

D  :  /x  [^  (Tp,  X)]v.'^\v.D.R€  V     (1 ) 

[r431]  t-:Hp.Pe/i.i;{^(Tp,X.)j/i.D.Pei;  (2) 

h  .  (1)  .  (2) .   D  h  :.  Hp  .  D  .  P  e  i"^  P^'X  :  D  :  /^  e  {Tp^Ayx  . 

^M.Pe/^     (3) 
h  .  (3)  .  D  h  .  Prop 

*-433.     h  :  Hp *-42  .  p  =  C'P^^  nR(R[ a'PQ'X  =  PQ'X) . 

D.'^(P<0'/3  =  tl(P'^)*X 
Dem. 

h  ::  Hpr42  .  D  :.  SeC'P'^' :  D  :  S[a'PQ'X  =  PQ'X  . 

y.'^z.ze a'PQ'X  .  ^ ['Q'z  =  PQ'X [ Q'z . 
(S'z)P{PQ'X'z)  .  V  .  (PQ'X'z)P(S'z) :. 


230    Miss  Wrinch,  On  the  exponentiation  of  well-ordered  series 

[-r4]     D  I- ::  Hp  .  D  :.  SeCP"^  .D  .  S  e  p  :v:  T  e  p  .Dj  ■ 

SP'iTiv.'g^U.UeX.UP^S: 

D  h  ::  Hp  .  D  :.  T^eX  .  D,- .  SP'^V :  S  eC'P'^ : 

DiSep.v.Tep.Dr-^'^P'^T: 


D  I- ::  Hp  .  D  :.  (T  =  ?7(FeX  .  D,- .  UP'W) -  max  (P'iyp  . 
SecTzD-.El  max  (P'^Yp  .  D  .  SP^  (max  (P'^Yp) . 

max  (P«)  V  epiv:  i^'P«  ^  =  A  .  D  .  gT .  .ST'-T .  T  e  p  : : 

D\-::lIl).D:.(T  =  U{Ve\.  D,- .  ?7P«F)  -  ^{P^p  . 

D.aCP'^"p     (1) 

h.(l).*205-193.        DI-:.Hp.D:(7  =  0'(FeX.D,--  t^P'^F) 

-l'^(P«)V  .  D  .  ^(P«)'p  u  cr  =  ^(P'?)'p      (2) 

h  .  *206-02  .  D  I- :.  Hp  .  D  :  prec  (P«)^X 

=  ^'  &(  F 6  /,; .  D  .  FP'^^  U)     (3) 

h  .  (3) .  :)  h  :.  Hp  .  D  :  (7  =  ^( Fe X  .  Dr  ■  ^i"^'^^) 

-  ^  (P«)'p  .  D  .  ^  (P«)'X  =  1^  (P'-')  Cp  yj  a)  : 

|-.(2).  Dh:.Hp.D:c7-f/(F6X.Dp..  f/^P^F) 

-  rmJ  (P«)>  .  D  .  prec  (P<^)'X  =  nmJ  (P''*)'p     (4) 

I- .  *-432  .  D  I- :  Hp .  D  .  i^  (P«)'X  =  A  (5) 

h  .  (4) .  (5) .  *207-02  .  D  h  :  Hp  .  D  .  ^  (P^Yp  =  tl  (P^)'^ 
r44.       h  :  Hp r42  . p  =  C'P^  nR{R\- d'PQ'X  =  PQ'X)  .pel. 

D  .  T'p  =  tl  (P«)'\     [*205-lS  .  *-433] 

*-45.       h  : .  Hp  r42  .  p  -  C"P«  n  P  (P  p  d'PQ'X  =  PQ'X) : 

D  :  a'PQ'X  =G'Q.  =  .p  =  I'PQ'X 
Dem. 

h  ::  Hp  .  D  :.  a'PQ'X  =  C'Q  .  D  .  PQ'X  e  G'P^  : 

Rep.D.a'R  =  a'PQ'X:: 

D  I- :.  Hp  .  D  :  a'PQ'X  =  C'Q  .  D  .  PQ'X  =  ^'p         (1) 
h  :.  Hp  .  D  :  /^  =  t'PQ'X  .  D  .  PQ'X  e  (7'P«  :. 


Miss  Wrinch,  On  the  exponentiation  of  well-ordered  series    231 

[*176-19]     D  f- :.  Hp  .  D  :  /?  =  i'PQ'X .  D  .  G'PQ'X  ■-=  C'Q         (2) 
l-.(l).(2).DI-.Prop 

*-451.     t-:.P,QeO.\CC*P«.X-'eOul  .p\Tp^Ay\  =  A. 

D  .  a'PQ'X  =C'Q:D.E !  tl  (P'?)'\     [**45-44] 

r46.        f- :,  P,  Q  e  a  .  \  C  0'P«  .  \  ~  e  0  u  1  .  p'(Tp*Ay\  =  A  . 

D^  .  E!  limin  (P^Yfi     [*rll-21 8-451] 

r5.         l-.Hpr42.D.maJ<2^a'PQ'X  =  A 

Pe?u. 
[*-04]  h.  Hp.^  6  a*PQ*\. 

D.'^fX,.fl€(Tp^AyX.2Q{Qm'f^)      (1) 

[r04]  h.Hp.yLie(rp*^yx. 

D.ai..^  =  TpV.Q'QmV.ca'PQ'x   (2) 

1- .  r21G  .  (1) .  (2) .  D  I- .  Hp .  ^  e  a'PQ'X . 

D.'^z'.zQz'.z'ea'PQ'X     (3) 

I- .  (3) .  D  h  .  Prop 

r51.       l-:.Hpr42:D:a*PQ'\=C'Q. 

V .  a^ .  ^  e  c"Q  -  a'Q, .  a'PQ'X = Q'^   [*-5] 

-r52.       \-:.P,Qen.C'QC  Q'Q, .  D  :  ya  ~  e  0  .  /^  C  G'P^  . 

D^  .  E  !  limin  (P«)V     [*r46-51] 

r53.       h  :  P,  Q  e  n  .  C'Q  C  Q'Q,  .  E  !  P'Cnv'P«  .  D  .  P<^'  e  Ded 

[**-52-l] 

r531.     \-:P,Qea.C'QC  a'Q,  .~B'Cnv'P'i  =  A  . 

D  .  P«  e  semi-Ded     [**-52-l] 

r5401.   h  :.  P,  Qe  n  :  D  :  p  =  B'Cnv'P^  .^.Rep. 

■D,.I)'R  =  i'B'P     [*17ryl9] 

*'541.     f- :.  P,  Q  e  O  .  D  :  E  !  5'Cnv'P«  .  =  .  E  !  5'P  [r5401] 

*-55.       h  : .  P,  Q  e  n  .  (7'Q  C  a^Qi :  D  :  P«  e  semi-Ded  : 

P«  e  Ded  .  =  .  E  !  B'P     [**-541-53-531] 
*-56.       h  : .  P,  Q  e  n  .  Nr'Q  ^  «  .  D  :  P«  e  semi-Ded  : 

P«eDed.=  .E!7i'P     [r55] 


232    Miss  Wrinch,  On  the  exponentiation  of  tuell-ordered  series 

*-6.         h  : .  Hp  *-42  .  p  =  C'P'^  r^  R[E\  a'PQ'X  =  PQ'X] .  p  ~  e  1  : 

D  :  E  !  B'P  .  =  .  E  !  max  (P'^'Yp 
Dem. 
[*176-19]         I-  :.  Hp  .  D  :R\a'PQ'\  =  PQ'X  .  Q'E  =  C'Q  . 

B'{B\-  a'PQ'X)  =  i'B'P  .  =^  .  E  -  max  (P^O^     (^) 

t- .  (1) .         D  h  :.  Hp  .  D  :  E  !  B'P  .  =  .  E  !  max  (P'O'p 
r61.       h:.P,Qe-Q.ao. 

\  =  P  (as  .  sQ^a  .  P  =  //1pT"Q'^  e;  (/2p  t^Q*'^)  • 

a  €  C'Q,  -  a'Qi :  D  :  E  !  limin  (P'-')'^  .  =  .  E  !  B'P 
Dem. 

h:.Hp:D:-(aP,AS.P,,S'6X.P4=6'.P[^"4.'a 

h  :.  Hp  .  D  :  ya  6  {Tp^AYX  .  D  .  Q^iV  Q  «  (1) 

h.(l).r04.  Dh:.Hp.D.a'PQ^\c'$a  .    (2) 

I- .  (2) .  D  h  :  Hp  .  D  .  p  -  e  1  (3) 

h  .  **-6-433  .    D  h  :.  Hp  .  D  :  E  !  limin  (P«)'\  .  s  .  E  !  P*P 
r62.       V:.P,QeVL.r^\G'Q-  Q'Qi .  D  -  ^  C  C"P«  .  X  -  e  0  : 

D  :  E  !  limin  {P'^yx  .  =  .  E  !  P'P     [^Ol] 

-r63.       h  :.  P,  Q  6 O  .  Nr'Q  >  «  :  D  :  P«  e  Ded  .  =  .  E  !  B'P  . 

B'P  =  A.  =  .  P«~esemi-Ded     [*r61-l] 

r7.         h  :.  P,  Q  6  O  .  D  :.  P«  6  Ded  .  =  .  E  !  P'P  [**-56-63] 

r8.         h  ::  P,  Q  6  O  .  D  :.  P«e  semi-Ded  .  =  .  E!  P'P  :  =  :  Nr'Q >  co 

Dem. 
h.*r63-7.  Dh::P,Qefl.D:.Nr'Q>ft): 

D  :  P«  e  semi-Ded  .  =  .  E  !  B'P     (1) 
|-.r56.        Dh::P,Q6f2.D:.Nr'Q^ft). 

D:P«esemi-Ded:E!P^P.v.p'^P  =  A     (2) 
|-.(1).(2).  DI-::P,Qeft:.Nr'Q>a): 

=  :  P«  e  semi-Ded  .  =  .  E  !  B'P 


Miss  Wrinch,  On  the  exponentiation  of  luell-ordered  series    233 

The  definitions  and  method  used  in  the  earlier  part  of  this  paper 
(**"01 — •341)  are  suggested  in  Principia  Mathematica  *27G.  There 
it  is  stated  tentatively  that 

g!  p'iT^Ayx  .  D  .  ^YiTp^Ayx  =  min  (P«)'X 
<-  g! p\Tp^Ayx  .  D  .  PQ'X  =  prec  (P^YX 

The  first  of  these  propositions  is  established  in  ***1 — •218 :  the 
second  seems  to  be  untrue.  If  in  the  field  of  Q  there  is  a  term  a 
with  no  immediate  predecessor  (as  for  example  the  term  co  if  Q 
were  the  series  of  ordinals  less  than  <w  +  4),  there  is  a  X,  a  subclass 
of  the  field  of  P***,  for  which  PQ'X  is  a  relation  covering  with  P-terms 
only  the  Q-terms  which  precede  a  (cp.  *"61).  In  such  a  case  PQ'X 
is  not  a  P*^  term  and  so  is  not  prec  (P^yx.  If  P  has  a  last  term  z, 
the  relation  agreeing  with  PQ'X  as  far  as  a  and  covering  a  and 
all  subsequent  Q  places  with  z  will  be  prec  {P'^yx,  and  therefore 
the  lower  limit  of  X  with  respect  to  P^. 

Thus,  while  agreeing  with  the  proposition  if  P  and  Q  are  well- 
ordered  series  and  P  has  a  last  term,  P^  is  Dedekindian,  and  ex- 
tending it  to  the  proposition  if  P  and  Q  are  well-orde7-ed  series, 
P'^  is  Dedekindian  tuhen  and  only  when  P  has  a  last  term,  we  dis- 
agree with  the  conclusion  that  if  P  and  Q  are  well-ordered  series, 
P'^  with  the  addition  of  a  term  at  the  end  is  Dedekindian  even  if  P 
has  no  last  term.  Instead  we  would  substitute  the  propositions 
when  P  and  Q  are  luell-ordered  series,  and  Nr'Q  ^  w,  P^  with  the 
addition  of  a  term  at  the  end  is  DedekiJidian  whetJier  or  not  P  has 
a  last  term,  and  if  Nr'Q  •>  o),  P^  with  the  addition  of  a  term  at 
tJie  end  is  Dedekindian  luhen  and  only  when  P  has  a  last  term. 


234  Mr  Neville,  The  Gauss-Bonnet  Theorem 


^ 


The  Gfinss-Bonnet  Tlieorem  for  Multiply -Connected  Jler/ions  of 
a  Surface.    By  Eric  H.  Neville,  M.A.,  Trinity  College. 

[Received  1  Dec.  1918:  read  8  Feb.  1919.] 

Among  the  most  delightful  passages  of  differential  geometry  is 
the  use  of  Green's  theorem  to  prove  the  relation  discovered  by 
Bonnet  between  the  integral  curvature  of  a  bounded  region  on 
any  bifacial  surface  and  the  integrated  geodesic  curvature  of  the 
boundary.    The  fundamental  equation  is 

,ds+l\Kd^S=  I'^^ds, 

.V  'as 

where  the  line  integrals  are  taken  round  the  whole  boundary  and 
the  surface  integral  over  the  region  contained,  Kg  is  the  geodesic 
curvature  of  the  boundary,  K  the  Gaussian  curvature  of  the 
surface,  and  f  an  angle  to  the  direction  of  the  boundary  from  the 
direction  of  one  of  the  curves  of  reference.  Though  there  is  no 
allusion  to  curves  of  reference  on  the  left  of  this  equation,  not 
only  do  these  curves  appear  explicitly  on  the  right,  but  the  use 
of  Green's  theorem  implies  that  there  does  exist  some  system  of 
curvilinear  coordinates  valid  throughout  the  region  and  upon  the 
boundary,  an  assumption  of  which  it  is  difficult  to  gauge  the  exact 
force.  The  primary  object  of  this  note  is  to  express  Bonnet's  theorem 
in  a  form  purely  intrinsic. 

In  the  case  of  a  simply-connected  region  not  extending  to 
infinity,  whose  boundary  has  continuous  curvature  at  every  point, 
the  value  of  J(d^/ds)ds  is  27r*.  If  the  region  is  simply-connected 
and  does  not  extend  to  infinity,  but  the  boundary  is  a  curvilinear 
polygon,  formed  of  a  finite  number  of  arcs  of  continuous  curvature, 
the  sum  of  the  external  angles  must  be  added  to  the  integral  to 
make  the  total  of  27r ;  in  other  words,  j{d^/ds)  ds  is  then  the 
amount  by  which  the  sum  of  the  external  angles  falls  short  of  27r. 
In  the  particular  case  of  a  curvilinear  triangle,  the  amount  by 
which  the  sum  of  the  three  external  angles  fails  short  of  'Itt  is  the 
amount  by  which  the  sum  of  the  three  internal  angles  exceeds  tt, 
and  is  called  the  angidar  excess  of  the  triangle.  The  name  is 
adopted  to  serve  a  wider  purpose :  whether  a  connected  region  of 
a  surface  is  bounded  by  a  single  closed  curve  or  by  a  number  of 

*  See  a  paper  by  G.  N.  Watson,  "A  Problem  of  Analysis  Situs"',  Froc.  Loud. 
Math.  Soc,  ser.  2,  vol.  15,  p.  227  (1916). 


for  Multiply-Connected  Regions  of  a  Surface  235 

curves,  the  amount  by  which  the  sum  of  all  the  external  angles 
of  the  boundary  falls  short  of  27r  is  called  the  angular  excess  of  the 
boundary. 

Whatever  the  number  of  curves  forming  the  boundary  of  a 
region,  the  addition  to  the  boundary  of  a  simple  cut,  joining  a 
point  of  the  boundary  either  to  a  point  of  the  cut  or  to  a  point  of 
the  boundary  and  described  once  in  each  direction,  increases  the 
sum  of  the  external  angles  by  27r.  If  the  cut  divides  the  region 
into  two  parts,  the  angular  excess  of  each  part  is  the  amount  by 
which  the  sum  of  the  external  angles  of  that  part  tails  short  of  27r, 
and  therefore  the  sum  of  the  two  angular  excesses  is  the  amount 
by  which  the  sum  of  the  external  angles  of  the  composite  boundary 
falls  short  of  47r;  this,  being  as  we  have  just  seen  the  amount  by 
which  the  sum  of  the  external  angles  of  the  original  boundary  falls 
short  of  27r,  is  the  angular  excess  of  the  original  boundary.  If  on 
the  other  hand  the  cut  leaves  the  region  undivided,  there  is  an 
actual  decrease  of  27r  in  the  excess.  It  follows  that  if  by  a 
succession  of  n  simple  cuts  the  region  is  divided  into  m  distinct 
parts,  the  sum  of  the  angular  excesses  of  the  boundaries  of  the 
parts  is  less  than  the  angular  excess  of  the  original  boundary  by 
2  (?i  —  in  +  1)  IT.  Suppose  now  that  each  of  these  parts  is  simpl}^- 
connected  and  that  there  are  no  singular  points  of  the  surface  ni 
the  original  region  or  upon  its  boundary.  Then  since  Bonnet's 
theorem  in  its  simplest  form  is  applicable  to  each  of  the  parts, 
addition  of  the  sum  of  the  integral  curvatures  of  the  parts  to  the 
sum  of  the  integral  geodesic  curvatures  of  the  boundaries  of  these 
parts  gives  the  sum  of  the  angular  excesses  of  the  individual 
boundaries.  But  the  sum  of  the  integral  curvatures  of  the  parts 
is  the  integral  curvature  of  the  original  region,  and  the  sum  of  the 
integral  geodesic  curvatures  of  the  boundaries  of  the  parts  is  the 
integral  geodesic  curvature  of  the  original  boundary,  since  an  arc 
described  once  in  each  direction  adds  nothing  to  JKgds.  Hence 
the  sum  of  the  integral  geodesic  curvature  of  the  original  boundary 
and  the  integral  curvature  of  the  bounded  region  is  less  than  the 
angular  excess  of  the  original  boundary  by  2  {n  —  ni  +  1)  tt.  This 
result  affords  a  proof  that  if  only  the  dissection  has  reached  a  stage 
at  which  every  part  is  simply-connected,  the  difference  n  —  m  is 
independent  alike  of  the  form  of  the  cuts  and  of  their  number. 
Since  a  simply-connected  region  is  divided  by  one  cut  into  two 
pieces,  the  integer  used  to  measure  connectivity  is  not  n  —  m  but 
n  —  in  -h  2,  and  Bonnet's  theorem  in  its  most  general  form  asserts 
that 

If  a  bounded  bifacial  region  of  any  surface  has  finite  con- 
nectivity k  and  neither  extends  to  infinity  nor  includes  tvithin  it  or 
upon  its  boundary  any  singularities  of  the  surface,  the  sum  of  the 
integral  geodesic  curvature  of  the  boundary  and  the  integral  curva- 


236  Mr  Neville,  The  Gauss-Bonnet  Theorem 


I 


ture  of  the  region  bounded  is  less  than  tlie  angular  excess  of  the 
boundary  by  2(^'  —  1)  ir. 

In  other  words,  the  sum  of  the  two  integrals  and  the  external 
angles  of  the  boundary  is  2  (2  —  k)  -k. 

Gauss'  famous  theorem  on  the  integral  curvature  of  a  geodesic 
triangle,  which  may  be  regarded  either  as  the  simplest  case  or  as 
the  ultimate  basis  of  Bonnet's  theorem,  is  in  no  less  need  of  modi- 
fication if  the  region  contemplated  is  multiply-connected. 

If  a  geodesic  triangle  on  any  surface  has  internal  angles  A,  B,C 
and  connectivity  k,  and  if  the  surface  is  regidar  throughout  the 
triangle  and  on  its  perimeter,  the  integral  curvature  of  the  triangle 
is  A  +  B-\-G-{'2.k-l)7r. 

The  application  to  the  whole  of  a  surface  which,  like  a  sphere 
and  an  anchor- ring,  does  not  extend  to  infinity,  but  has  no 
boundary,  is  interesting.  A  simple  closed  curve  can  always  be 
drawn  to  divide  such  a  surface  into  two  distinct  parts,  and  since 
its  direction  as  the  boundary  of  one  part  is  opposite  to  its  direction 
as  the  boundary  of  the  other  part,  the  sum  of  the  external  angles  of 
the  two  boundaries  is  zero,  and  so  also  is  the  sum  of  their  integral 
geodesic  curvatures.  It  follows  from  Bonnet's  theorem  that,  if 
there  are  no  singular  points  on  the  surface  and  the  connectivities 
of  the  two  parts  are  i,  j,  the  integral  curvature  of  the  complete 
surface  is  2  (4  —  i  —j)  it.  Hence  i  +j  is  constant ;  in  order  that  a 
surface  which,  like  a  sphere,  is  cut  by  any  simple  closed  curve  into 
two  simply-connected  parts  may  be  described  as  of  unit  con- 
nectivity, the  connectivity  is  measured  by  the  integer  i+j—1, 
and 

If  the  connectivity  of  a  bifacial  surface  which  has  no  boundary 
and  no  singular  points  and  does  not  extend  to  infinity  is  k,  the 
integral  curvature  of  the  surface  is  2  (3  —  k)  ir. 

A  striking  deduction  made  by  Darboux  from  Bonnet's  theorem 
may  be  mentioned  here.  If  on  a  complete  surface  there  is  any 
family  of  curves  such  that  the  surface  can  be  divided  into  a  finite 
number  of  parts  throughout  each  of  which  this  family  provides 
one  set  of  curves  of  reference,  the  angle  ^  of  our  first  paragraph 
can  be  measured  from  the  curve  belonging  to  this  family,  and 
J{d^/ds)ds  taken  once  in  each  direction  over  every  part  of  an 
imposed  boundary  is  necessarily  zero.    Hence 

For  there  to  exist  on  an  unbounded  bifacial  surface,  which  does 
not  extend  to  infinity  and  is  everywhere  regidar,  afainily  of  curves 
which  covers  the  surface  and  is  wholly  withoid  singularities,  the 
surface  must  have  integral  curvature  zero  and  must  therefore  be 
triply-connected. 

In  conclusion  the  subject  may  be  presented  in  another  form. 
Let  the  angular  excess  of  the  boundary  of  a  region  of  connectivity 
k  reduced  by  2(A;-  l)7r  be  called  the  effective  angular  excess.    If 


for  Multiply-Connected  Regions  of  a  Surface      ^       237 

a  simple  cut  which  is  added  to  the  boundary  does  nut  divide;  the 
region,  the  anguh^r  excess  is  reduced  by  27r,  and,  since  the  con- 
nectivity is  reduced  by  unity,  the  etfective  angular  excess  is 
unaltered.  If,  on  the  other  hand,  the  cut  divides  the  region  into 
parts  of  connectivities  i,  j,  not  only  is  the  sum  of  the  actual  angular 
excesses  of  the  boundaries  of  the  parts  the  actual  angular  excess 
of  the  original  boundary,  but,  since  /;  is  i  +j  —  1,  the  sum  of  i  —  1 
and  J  —  1  is  ^'  —  1 :  the  effective  angular  excess  of  the  boundary  of 
the  whole  is  the  sum  of  the  effective  angular  excesses  of  the 
boundaries  of  the  parts.  Effective  angular  excess  is  therefore 
additive  in  precisely  the  same  way  as  the  surface  integral  of  a 
single-valued  function.  If  then  Bonnet's  theorem  for  a  simply- 
connected  region  is  expressed  in  the  form  that  the  sum  of  the 
integral  curvature  and  the  integral  geodesic  curvature  is  the 
effective  angular  excess,  the  restriction  on  the  connectivity  is  seen 
at  once  to  be  superfluous.  But  to  take  this  course  implies  a 
previous  acquaintance  with  the  theory  of  connectivity,  whereas  it 
is  arguable  that  if  Bonnet's  theorem  is  used  to  establish  the  theory 
of  connectivity  the  extent  to  which  there  is  an  appeal  to  intuition 
is  materially  reduced. 


238    Mr  8hah  and  Mr  Wilson,  On  an  empirical  formula,  connected 

On  an  empirical  formula  connected  with  GoldhacJis  Theorem. 
By  N.  M.  Shah,  Trinity  College,  and  B.  M.  Wilson,  Trinity  Col- 
lege.   (Communicated  by  Mr  G.  H.  Hardy.) 

[Received  20  January  1919 :  read  3  February  1919.] 

§  1.  The  following  calculations  originated  in  a  request  recently 
made  to  us  by  Messrs  G.  H.  Hardy  and  J.  E.  Littlewood,  that  we 
should  check  a  suggested  asymptotic  formula  for  the  number  of 
ways  V  (n)  of  expressing  a  given  even  number  n  as  the  sum  of  two 
primes.    The  formula  in  question  is 

vin)^\{n)  =  2Aj^^/^l^^^ (1), 

^  ^         ^  ^  (log ny p-2q  —  2  ^  ^ 

where  ??,  =  2'^p«^^ ...     (a^l) 

and  A  denotes  the  constant 

CO   J-  I 

p  assuming,  in  this  product,  the  odd  prime  values  3,  5.  7, 11, 13,  .... 
The  formula  (1)  was  deduced  from  another  conjectured  asymp- 
totic formula,  namely 

X      A{m)A(m')^2An^^^ (2), 

where  A  (m)  is  the  arithmetical  function  equal  to  log  p  when  m  is 
a  prime  p,  or  a  power  oi  p,  and  to  zero  otherwise,  and  the  summation 
on  the  left  is  extended  to  all  pairs  of  positive  integers  m,  m'  such 
that 

m  +  m  =  n. 

Formula  (1)  arises  from  (2)  by  replacing  in  the  latter  A{m)  and 
A {m)  each  by  log n.  It  is  natural,  however,  to  expect  a  more 
accurate  result  if  we  replace  A  (m)  and  A  {ni)  not  by  log  n  but  by 
\og^n,  or,  better  still,  if  we  replace  the  left-hand  member  of  (2)  by 

— ^       log  X  log  {n  —  x)dx (3). 

1^       .  0 

The  exact  value  of  the  expression  (3)  is  found  to  be 

V  (n)  {(log  ny  -  2  log  n  +  2-  i-tt^}     (4). 

The  various  formulae  thus  obtained  from  (2)  are,  of  course,  all 
asymptotically  equivalent ;  but  the  modified  formulae  are  likely  to 
give  more  accurate  results  than  (1)  for  comparatively  small  values 
of  n.    We  used  the  formula 

v(n)^p(n)=2A-. -^. ^^^~i (5), 

^  ^      •   ^  ^  (log ny-2\ogn p  —  2  q-2  ^  ^ 

obtained  by  ignoring  the  constant  2  —  ^tt'^  in  (4). 


luith  Goldhach's  Theorem  239 

§  2.  For  the  numerical  data  used  we  are  indebted  to  two 
different  sources.  The  most  complete  numerical  results  are  con- 
tained in  the  tables  compiled  and  published*  by  R.  Haussner,  which 
give  the  values  of  v(n)  for  all  values  of  n  not  exceeding  5000. 
Tables  extending  up  to  1000  and  2000  had  been  calculated  earlier 
by  G.  Cantor  and  V.  Aubry.  Further  data,  less  systematic,  indeed, 
than  those  of  Haussner,  but  extending  to  considerably  larger  values 
of  n,  were  given  by  L.  Ripertf  in  a  number  of  short  papers  in 
V Intermediaire  des  mathe'inaticiens. 

The  values  given  for  v()i)  in  the  accompanying  table  differ,  in 
several  respects,  from  those  given  by  Haussner  or  Ripert.  In  the 
first  place,  7n  +  m  and  m  +  m  are  here  counted  as  different  decom- 
positions, whereas  the  above  two  writers  regard  them  as  identical ; 
secondly  we  do  not  (as  do  Haussner  and  Ripert)  regard  1  as  a 
prime ;  and  thirdly  we  increase  the  values  of  v  (n)  obtained  from 
their  tables  by  addition  of  the  number  of  ways  in  which  n  may  be 
expressed  as  the  sum  of  two  powers  of  primes,  i.e.  the  number  of 
ways  in  which 

n=p'^  +  (f', 
where  j)  and  q  are  primes,  and  either  a  or  b  is  greater  than  unity. 
The  last  two  modifications  make,  of  course,  no  difference  to  the 
asymptotic  formula,  but  it  seems  natural  to  make  them  when  the 
genesis  of  the  formula  (1)  or  (5)  is  considered. 

As  regards  the  choice  and  arrangement  of  the  numbers  n  in  the 
table,  the  smaller  numbers — i.e.  the  numbers  not  exceeding  5000 
— are  intended  to  be  "  typical  "  ;  that  is,  they  are  specially  selected 
numbers,  taken  in  groups  so  as  best  to  test  or  illustrate  the  accuracy 
of  formula  (1).  Thus,  for  example,  if  the  formula  in  question  is  true, 
a  multiple  of  6  may  be  expected,  in  general,  to  allow  of  an  unusually 
large  number  of  decompositions :|:.  On  the  other  hand  a  power  of  2 
may  be  expected  to  allow  of  an  unusually  small  number.  The 
numbers  below  5000  have  therefore  been  selected  in  groups  of  four 
or  five,  all  the  numbers  of  each  group  being  as  nearly  equal  as 
possible ;  and  each  group  of  numbers  contains,  in  general,  one 
highly  composite  number  (i.e.  2.3.5.7,11....),  one  power  of  2, 
and  one  number  which  is  the  product  of  2  and  a  prime. 

For  values  of  n  exceeding  5000,  such  choice  of"  typical "  numbers 
was,  unfortunately,  impossible  without  a  large  amount  of  fresh 
calculation.  Ripert,  indeed,  selected  his  numbers  according  to  a 
system,  and  they,  too,  occur,  in  general,  in  gToujJs  of  approximately 
equal  magnitude;  but  he  selected  them  with  different  objects,  so 
that  his  numbers  are,  from  our  point  of  view,  neither  "  typical  "  nor 
arbitrary. 

*  Nova  Acta  tier  Akad.  der  Natur/orscher  (Halle),  vol.  72  (1897),  pp.  5-214. 
t  See,  for  example,  vol.  10  (1903),  pp.  76-77,  16(3-167. 

X  It  was  first  pointed  out  by  Cantor,  on  the  evidence  of  his  numerical  results 
previously  mentioned,  that  this  is  actually  so. 

VOL.   XIX.   PART  V.  17 


240    Mr  Shah  and  Mr  Wilson,  On  an  empirical  formula  connected 

The  accompanying  table  gives  the  number  of  decompositions — 
actual  and  theoretical — for  thirty-five  numbers  ;  the  value  found 
for  the  constant  A  was  0"66016.  In  the  second  column  the  first 
number  is  the  number  of  decompositions,  using  prime  numbers  only, 
and  the  second  the  number  of  decompositions  involving  powers  of 
primes  higher  than  the  first. 


§  3.     Table  of  decompositions. 

n 

V  («) 

P  («) 

v{n):p{n) 

30  =  2.3.5 

32  =  2" 
34  =  2.17 
36  =  22.32 

6+  4=      10 

4+   7=      11 
7+   6=      13 
8+   8=      16 

22 
8 
9 

17 

•45... 
1-38... 
1-44... 

•94 

210  =  2.3.5.7 
214  =  2.107 
216  =  23.33 
256  =  28 

42+   0=      42 
17+  0=      17 
28+   0=       28 
16+   3=      19 

49 
16 
32 
17 

•85 
1-07 

•88 
1-10 

2,048  =  211 
2,250  =  2.32.53 
2,304  =  28.32 
2,306  =  2.1153 
2,310  =  2.3.5.7.11 

50  +  17=      67 
174  +  26=    200 
134+   8=     142 

67  +  20=      87 
228  +  16=    244 

63 
179 
136 

69 

244 

1-06 
1^11 
1^04 
r26 
1-00 

3,888  =  2*.  35 
3,898  =  2.1949 
3,990  =  2.3.5.7.19 
4,096  =  212 

186  +  24=    210 

99+   6=    105 

328  +  20=    348 

104+   5=     109 

197 

99 

342 

102 

1-06 

ro6 

1^02 
1-06 

4,996  =  22.1249 
4,998  =  2.3.72.17 
5,000  =  23.5* 

124  +  16=     140 
288  +  20=    308 
150  +  26=    176 

119 
305 
157 

1-18 

roi 

1-12 

8,190  =  2.32.5.7.13 

8,192  =  213 

8,194  =  2.17.241 

578  +  26=    604 
150  +  32=     182 
192  +  10=    202 

597 
171 
219 

1^01 

roe 

•92 

10,008  =  23.32.139 
10,010=2.5.7.11.13 
10,014  =  2.3.1669 

388  +  30=    418 
384  +  36=    420 
408+   8=    416 

396 

384 
396 

1^06 
1^09 
1^05 

30,030  =  2.3.6.7.11.13 
36,960  =  25.3.5.7.11 
39,270  =  2.3.5.7.11.17 

41,580=22.33.5.7.11 

1,800  +  54  =  1,854 
1,956  +  38  =  1,994 
2,152  +  36  =  2,188 

2,140  +  44  =  2,184 

1,795 
1,937 

2,213 
2,125 

1-03 

1^03 

•99 

1-03 

50,026  =  2.25013 
50,144  =  25.1567 

702+   8=     710 
674  +  32=    706 

692 
694 

1^03 

ro2 

170,166  =  2.3.79.359 
170,170  =  2.5.7.11.13.17 
170,172  =  22.32.29.163 

3,734  +  46  =  3,780 
3,784+   8  =  3,792 
3,732  +  48  =  3,780 

3,762 

3,841 
3,866 

1^00 
•99 

•98 

ivith  Goldbach's  Theorem  241 

§4.  Goldbach  asserted  that  every  even  number  is  the  sum  of 
two  pnmes,  and  this  unproved  proposition  is  usually  called  'Gold- 
bach  s  Theorem'.  It  is  evident  that  the  truth  of  Hardy  and 
Littlewood's  formula  would  imply  that  of  Goldbach's  theorem,  at 
any  rate  for  all  numbers  from  a  certain  point  onwards. 

Previous  writers,  from  Cantor  onwards,  had  noted  that  the 
HTegularity  m  the  variation  of  j/(«)  depends  on  the  structure  of  n 
as  a  product  of  pi-imes.  In  a  short  abstract  in  the  Proceedings  of 
the  London  Mathematical  Society,  Sylvester*  suggested  the  formula 

,  .        2n       p-2 
'^'^-logn^p^   (6), 

where,  in  the  product  on  the  right  p  assumes  all  prime  values  from 
3  to  Vw,  except  those  which  are  factors  of  n.  Sylvester  gives  but 
little  indication  as  to  how  he  arrived  at  the  formula  and  indeed 
there  is  much  m  his  paper  which  is  not  very  clear.  It  is  at  once 
obvious  that  if  71,  n'  are  two  large,  but  approximately  equal  even 
numbers,  the  values  furnished  for  the  ratio  v{n):v  {n')  by  formulae 
(1)  and  (6)  will  be  the  same.    For  if 

n  =2"-p'^  q^ ... 

and  n  =  2'^  p'^'  q'^ 

both  formulae  will  give,  as  an  approximate  expression  for  this  ratio 
the  quotient  ' 

p-2  q-2'"  I  p  -2  q  -2"" 

The  actual  values  of  v  {n)  would  however  be  different.  For  from 
formula  (6)  we  should  deduce 

\ognp-2q-2-j,'^lji-l- 

Now         n  ^-2=  n  Pil:z^ 


'A    U    (1-- 


p<\'n  \         PJ 

where  A  is  the  same  constant  as  in  formula  (1).    Also  it  is  known  f 


that 

IN       2e-y 


n     1- 

P<sJn\         p)        log/i 

^■,1  r  ^1°"-  f  °"^«"  ^^««''-  ^oc-.  vol.  4  (1871),  pp.  4-6  (il/atft.  Papers,  vol.  2,  pp  709- 
711).    See  also  Math.  Paj^ers,  vol.  4,  pp.  734-737.  >  I'r    «   c; 

t  Landau,  Handbuch  der  Lehre  vuii  der  Verteilung  der  Primzahlen,  p.  140. 

17—2 


242    Mr  Shah  and  Mr  Wilson,  On  an  empirical  formula  connected 


so  that  (6)  is  equivalent  to 

^  ^  (logw)^  j[9  —  2  g  — 2 

Hence  the  asymptotic  values  furnished  for  v{n)  by  (6)  and  by  (1) 
are  in  the  ratio  2e~'>'  :  1,  i.e.  in  the  ratio  1123  :  1. 

A  quite  different  formida  was  suggested  by  Stackel*,  viz. 

(log?i)2  0(n) 
where  ^  (n)  denotes,  as  usual,  the  number  of  numbers  less  than  n 
and  prime  to  n.    This  is  equivalent  to 


V  (n) ' 


P 


.(9). 


(log  ny  p  —  lq  —  1 

Since  p/(p—l)  is  nearer  to  unity  than  (p  — l)/(p  — 2),  the 
oscillations  of  v  (n)  would,  if  Stackel's  formula  were  correct,  be 
decidedly  less  pronounced  than  they  would  be  if  (1)  were  correct. 
As  between  the  two  formulae,  the  numerical  evidence  seems  to  be 
decisive.  Thus  the  ratio  z^(8190)  :  z/(8192)  is  3-32,  whereas  ac- 
cording to  (1)  it  should  be  3"48,  and  according  to  Stackel's  formula 
it  should  be  2*37.  Stackel's  i-esult  is  obtained  by  considerations  of 
probability  which  ignore  entirely  the  irregularity  of  the  distribution 
of  the  primes  in  a  given  interval  ii^N,  and  it  is  not  surprising, 
therefore,  that  it  should  be  seriously  in  error. 

On  the  other  hand  it  should  be  observed  that  Sylvester's  for- 
mula (7)  gives,  within  the  range  of  the  table  on  p.  240,  very  good 
results,  not  much  worse  than  those  given  by  (5),  and  decidedly 
better  than  those  given  by  (1).  This  is  shown  by  the  table  which 
follows,  in  which  decompositions  into  powers  of  primes  higher  than 
the  first  are  neglected. 


n 

Formula  (7) 
v{n):2e-y\{n) 

Formula  (1) 
v{n)  -.X  {n) 

2,048  =  2" 
2,250  =  2.32.53 
2,304=28.32 
2,306  =  2.1153 
2,310  =  2.3.5.  7.  11 

•95 
1-17 
1-18 
1-17 
1-12 

1-06 
1-31 
1-33 
1-31 
1-26 

10,008  =  23.32.139 
10,010  =  2.5.7.11.13 
10,014  =  2.3.1669 

1-11 
1-12 
1-17 

1-25 
1-27 
1-32 

170,166=2.3.79.359 
170,170  =  2.5.7.11.13.17 
170,172  =  22.32.29.163 

1-06 
1-05 
1-04 

1-19 
1-18 
1-16 

Gottinger  Nachrichten  (1896),  pp.  292-299. 


with  Goldbach's  Theorem  243 

§  5.     It  has  been  shown  by  Landau*  that 

Sv(/0~„-7r— ^  (10); 

1     ^  ^      2(logn)2  ^     ^' 

and  that  Stackel's  formula  (8)  is  inconsistent  with  (10),  and  ac- 
cordingly incorrect. 

The  same  test  can  be  applied  to  the  formula  (1)  and  Sylvester's 
formula  (7).  In  fact  Messrs  Hardy  and  Little  wood  have  shown  f 
that  (10)  is  a  consequence  of  (1)  :  from  which  it  follows,  of  course, 
that  the  asymptotic  formula  of  the  type  of  (10),  furnished  by 
Sylvester's  formula,  would  be  in  error  to  the  extent  of  a  factor 
2e~'>'=  ri23 ;  that  Sylvester's  formula  is  therefore  also  incorrect ; 
and  that  if  any  formula  of  this  type  is  correct,  it  must  be  (1). 

It  may  seem  at  first  surprising  that,  in  these  circumstances, 

Sylvester's  formula  should  give,  for  fairly  large  values  of  n,  results 

actually  better  (as  is  shown  by  the  results  in  the  table  on  p.  242) 

than  those  given  by  (1).     The  explanation  is  to  be  found  in  the 

nature  of  the  error  term  in  (1).     The  modified  formula  (5),  which 

we  have  already  shown  to  be  likely  to  give  better  results  than  (1), 

for  moderately  large  values  of  n,  differs  from  (1)  by  a  factor  of  the 

type  9 

1+r^  +.... 
logn 

This  factor  does  not  affect  the  asymptotic  value  of  v  (n),  but  it 
makes  a  great  deal  of  difference  within  the  limits  throughout 
which  verification  is  possible  :  thus  when  n=  170,170  it  is  equal  to 
1"166.  When  n=  10^",  it  is  equal  to  1"087,  and  its  difference  from 
unity  is  negligible  only  when  n  is  quite  outside  the  range  of 
computation.  It  is  only  such  values  of  n  that  would  reveal  the 
superiority  of  the  unmodified  formula  (1)  over  Sylvester's  formula. 

1 6.  Shortly  after  the  writing  of  the  preceding  sections  had  been 
completed,  Mr  Hardy  informed  us  of  the  existence  of  a  third  pro- 
posed asymptotic  formula  for  v  (n),  given  more  recently  by  V. 
Brun;|:.     The  formula  to  which  Brun's  argument  leads  is 

v{n)^2Bn^'~l^^ (11), 

p—z q—2 

where  5=  {  1  -  "  ]  ( 1  -  "  )  ( 1 -^  ) ...  ( 1 -r 


=(-!) 

('-?) 

(^ 

2' 

7, 

h<\'u    f 

=  n   (1 

-I)- 

7i  =  3     V 

"  GiHtinger  Nachrichten  (1900),  pp.  177-186. 
t  See  their  note  which  follows  this  paper. 

:!:  Archiv  for  Mathematik  (Christiauia),  vol.  34,  1917,  no.  8.     See  also  §  4  of 
Hardy  and  Littlewood's  note. 


244  Mr  Shah  and  il/?-  Wilson,  On  GoldbacJiS  Theorem 

By  an  argument  similar  to  that  used  in  §4,  in  the  reduction  of 
Sylvester's  formula,  it  may  be  shown  that  this  is  equivalent  to  the 
formula 

i.(n)~8^e-^y^  ,,^^...=4e-=yX(70     (12). 
(log nf  p-2  q-  2 

Thus  this  asymptotic  value  for  v  (n),  and  the  Hardy-LittleAvood 
value,  are  in  the  ratio  4e~-T  :  1  =  1'263...  :  1.  Sylvester's  is  their 
geometric  mean. 

The  formulae  (11)  and  (12)  would  furnish  a  quite  close  ap- 
proximation for  V  (n)  for  those  values  of  oi  on  which  it  could  be,  in 
practice,  tested.     Thus,  for  n  =  170,170,  we  find  that 

v{n)/4^e-'yX{n)  =  -9S.... 

But  the  ultimate  incorrectness  of  the  formula  may  be  proved  in 
the  same  way  as  that  of  Sylvester's  formula,  namely  by  use  of 
Landau's  asymptotic  formula  (10). 

Brun  knew  of  the  memoirs  of  Stackel  and  Landau,  but  appears 
to  have  been  unacquainted  with  Sylvester's  work. 


Mr  Hardy  and  Mr  Littlewood,  Note  on  Messrs  Shah,  etc.   245 


Note  on  Messrs  Shah  and  Wilson's  paper  entitled:  'On  an 
empirical  formula  connected  with  Ooldbach's  Theorem  '.  By  G.  H. 
Hardy,  M.A.,  Trinity  College,  and  J.  E.  Littlewood,  M.A., 
Trinity  College. 

[Received  22  January  1919:  read  3  February  1919.] 

1.  The  formulae  discussed  by  Messrs  Shah  and  Wilson  were 
obtained  in  the  course  of  a  series  of  researches  which  have  occupied 
us  at  various  times  during  the  last  two  years.  A  full  account  of 
our  method  will  appear  in  due  course  elsewhere*:  but  it  seems 
worth  while  to  give  here  some  indication  of  the  genesis  of  these 
particular  formulae,  and  others  of  the  same  character.  We  have 
added  a  few  words  about  various  questions  which  are  suggested  by 
Shah  and  Wilson's  discussion. 

The  genesis  of  the  formulae. 

2.  Let 

f{x)  =  %A  (n)  X''  =  SA  (n)  e-"y  =  F{ij) 

and  /,  (x)  =  F,  {y)  =  Ix^  (n)  A  (n)  e"".", 

where  A  (n)  is  equal  to  logp  when  n  is  a  prime  p,  or  a  power  of  ^j, 
and  to  zero  otherwise,  and  x<  (^0  i^  one  of  Dirichlet's  '  characters  to 
modulus  (/'+.    Also  let 

X  =  xe'^'^''?, 

where  p  is  positive,  less  than  q,  and  prime  to  q ;  and  suppose  that 
X  tends  to  unity  by  positive  values. 
It  is  known  that 

n 

^X'^{v)A{v)  =  o{n\ 
1 

unless  Xk  is  the  '  principal '  character  Xi>  ^^  which  case 

n  n 

1  1 

It  follows  that 

(2-1)  A(K)'^--  ^ 


l-x 

and 

(2-2)  f^^''^  =  ^{l^     ^'^^l^- 

*  An  outline  of  one  of  its  most  important  applications  is  contained  in  a  paper 
entitled  '  A  new  solution  of  Waring's  Problem  ',  which  will  be  ijublished  shortly  in 
the  Quarterly  Journal  of  Mathematics. 

t  See  Landau,  Haudbuch,  pp.  391  et  seq. 


■I 


246  Mr  Hardy  and  Mr  Littleivood,  Note  on 

Now 
(2-3)      /(^)  =  2A(?2)x"e2n/.'^^'9=  S  e=-^^^'^'"?   2  A(n)x>\ 

i  =  l  n=,i 

If  J  is  prime  to  q,  we  have* 
(2-4)  S   A  (n)  X-  =  -^  'S  X.  ( j)  /.  (X), 

where  %«  is  the  character  conjugate  to  %«,  and  ^(5')  is  the  number 
of  numbers  less  than  and  prime  to  q.  It  follows  from  (2'1)  and 
(2-2)  that 

(2-5)  S  A  (n)  x»  ^  144  r^-  =  X?-M  ^     • 

n=j  </)(g)  1-X        <l){q)l-K 

If  on  the  other  hand  j  is  not  prime  to  q,  the  formula  (2'4)  is 
untrue,  as  its  right-hand  side  is  zero.  But  in  this  case  A  (??)  =  0 
unless  n  is  a  power  of  q,  so  that 

(2-6)  S  Ain)K^^=^o(~). 

From  (2-3),  (2-5),  and  (2-6)  it  follows  that 

(2-7)  •^^^•)~l~x' 

where 

^  ^  '    <P{q)7  ^{q)7'     ' 

the  summation  extending  over  all  values  of  j  less  than  and  prime 
to  q.  The  sum  which  appears  in  (2*71)  has  been  evaluated  by 
Jensen  and  Ramanujanf,  and  its  value  is  /Li{q),  the  well-known 
arithmetical  function  of  q  which  is  equal  to  zero  unless  5  is  a  product 
JJ1P2  ■••  Pp  of  different  primes,  and  then  equal  to  (—  1)p.     Thus 


(^•«>  f^^>-mTh.t- 


3.     The  sum 
(3-1)  co{n)=      t      A(m)A(??0, 

■m  +  m'=n 
*  Landau,  I.e.,  p.  421. 

m 

t  J.L.W.V.  Jensen, 'EtnytUdtryk  for  den  talteoretiske  Funktion  2M(n)  =  M{my, 

1 
Saertryk  af  Beretning  om  den  3  Skandinaviske  Matematiker-Kongres,  Kristiania, 
1915 ;  S.  Eamanu jan,  '  On  certain  trigonometrical  sums  and  their  applications  in  the 
theory  of  numbers',  Trans.  Camb.  Phil.  Soc,  vol.  22,  1918,  pp.  259-276. 

J  If  ^  (q)  is  zero,  this  formula  is  to  be  interpreted  as  meaning 


/w=o(r^) 


Messrs  Shah  and  Wilsons  paper  247 

which  appears  on  the  left-hand  side  of  Shah  and  Wilson's  equation 
(2),  is  the  coefficient  of  a;'*  in  the  expansion  of  [/(«))".     And 

when  a;  ^  e2i?7ri7(?  along  a  radius  vector.  Our  general  method  ac- 
cordingly suggests  to  us  to  take 

n(n)  =  nt\^i(i^'e-'-^^P-il<i, 

where  the  summation  extends  over  5'=  1,  2,  3,  ...  and  all  values 
of  jt)  less  than  and  prime  to  q,  as  an  approximation  to  co  (n).  Using 
Ramanujan's  notation,  this  sum  may  be  written 

(3-2)  n{n)=nl\^^l^%,(n). 

The  series  (3'2)  can  be  summed  in  finite  terms.     We  have 

(3'3)  c,(7i)  =  SS/.(| 

the  summation  extending  over  all  common  divisors  S  of  q  and  n*; 
and  it  is  easily  verified,  either  by  means  of  this  formula  or  by  means 
of  the  definition  of  Cq{n)  as  a  trigonometrical  sum,  that 

Cqq'{n)  =  Cq{n)Cg'(n) 

whenever  q  and  q'  are  prime  to  one  another.  We  may  therefore 
write 

n(n)=n:ZAq  =  nUx-r!r, 
where  the  product  extends  over  all  primes  -or,  and 

since  Aq  contains  the  factor  /j,(q)  and  A^^^.,  A ^-3,  ...  are  accordingly 
zero. 

If  n  is  not  divisible  by  zj,  we  have  c^  (n)  =  /u,  (ot)  =  —  1  and 

A    =_^    1       = L__. 

while  if  n  is  divisible  by  zr  we  have 

Cw('0  =  At(t3-)  +  t3-/A(l)=  OT  -  1, 

.     _     1 

CT  —   1 

Hence 

*  Ramanujan,  I.e.,  p.  260. 


248  Mr  Hardy  and  Mr  Littlewood,  Note  on 

where   11'  applies  to   primes  which  divide  n  and  II"  to  primes 
which  do  not. 

It  is  evident  that  O  (n)  is  zero  if  n  is  odd.    On  the  other  hand, 
if  n  is  even,  we  have 


"W  =  2»n|i-(-'-3y,}n 


-^)/|-.^-:. 


=  2^nn-^^ln^P-^ 


where  tn-  now  runs  through  all  odd  primes  and  p  through  odd 
prime  divisors  of  n. 

The  formula  w  (n)  ^-  il  (n) 

is  formula  (2)  of  Shah  and  Wilson's  paper*. 

The  incorrectness  of  Sylvester's  formula. 

4.     It  is  easy  to  prove  that  if  any  form  tda  of  the  type 

(4-1)  fw  (7^)  ~  CO  (n) 

be  true,  then  G  must  be  unity.     In  other  words,  our  formula  is  the 
only  formula  of  this  type  which  can  possibly  be   correct.     This 
may  be  shown  as  follows. 
Let 

(4-2)  f(^)  =  t'l^l 

where  n  runs  through  all  even  values;  and  let  s  —  1  =  ^.  The  series 
is  absolutely  convergent  if  s  >  2,  ^  >  1.  Replacing  12  (n)  by  its 
expression  in  terms  of  the  prime  divisors  of  n,  and  splitting  up 
f{s)  into  factors  in  the  ordinary  manner,  we  obtain 

say,  where  A  is  the  same  constant  as  in  Shah  and  Wilson's  paper, 
and  OT  runs  through  all  odd  primes. 
Let 

+  (*)  =  n  (i  +  r^--.)  =  n  [^}^_)  =  (1  -  2-0  f  (0. 

and  suppose  that  ^^1.     Then 


x(0.nifi+^-4,*"4Vfi+i-^)} 


'y\r{t)  [\         OT-2  1- 


•37 


V(^-2)|      ^'1(37-1)^-11      A' 
When  fi(n)  =  0,  the  formula  is  to  be  interpreted  as  meaning  w(H)  =  o(/t). 


Messrs  Shah  and  Wilsons  paper  249 

and  so 

(4-3)       f{s)  ~  2^^  (0  -2(1-  2-0  ?(0  -  ^1  =  ,--2  • 

This  is  a  consequence  of  our  hypothesis :   the  corresponding 
consequence  of  the  hj^pothesis  (4"1)  would  be 

(4-31)  /(*^>~^- 

On  the  other  hand,  it  is  easy  to  prove*  that 
(4-4)  &)(l)  +  ft)(2)+ ... +&)(7i)~i«'; 

and  from  this  to  deduce  that 


<^(.)  =  2 


on 


(n)         1 


w*        s  —  2 

when  s—>2.  This  equation  is  inconsistent  with  (4"1)  and  (4'31), 
unless  (7  =  1. 

It  follows  that  Sylvester's  suggested  formula  is  definitely 
erroneous. 

It  is  more  difficult  to  make  a  definite  statement  about  the 
formula  given  by  Brun.  The  formula  to  which  his  argument 
naturally  leads  is  Shah  and  Wilson's  formula  (12);  and  this 
formula,  like  Sylvester's,  is  erroneous.  But  in  fact  Brun  never 
enunciates  this  formula  explicitly.  What  he  does  is  rather  to 
advance  reasons  for  supposing  that  some  formula  of  the  type  (4"1) 
is  true,  and  to  determine  G  on  the  ground  of  empirical  evidence^. 
The  result  to  which  be  is  led  is  equivalent  to  that  obtained  by 
taking  C=  1-5985/1-3203  =  1-2107  %.  The  reason  for  so  substantial 
a  discrepancy  is  in  effect  that  explained  in  the  last  section  of 
Shah  and  Wilson's  paper. 

Further  results. 

5.  The  method  of  §  2  leads  to  a  whole  series  of  results  con- 
cerning the  number  of  decompositions  of  n  into  3,  4,  or  any  number 
of  primes.     The  results  suggested  by  it  are  as  follows.     Suppose 

*  Since  SA  (71)  .r™-:; 

as  a;-*-l,  we  have  2w(H)a;"=  (SA  («)  a;"}-~  ,-    -  -', 

and  the  desired  result  follows  from  Theorem  8  of  a  paper  published  by  us  in  1912 
('  Tauberian  theorems  concerning  power  series  and  Dirichiet's  series  whose  coefficients 
are  positive',  Proc.  London  Math.  Soc,  ser.  2,  vol.  13,  pp.  174-192).  This,  though 
the  shortest,  is  by  no  means  the  simplest  proof. 

The  formula  (4-4)  is  substantially  equivalent  to  Landau's  formula  (10)  in  Shah 
and  Wilson's  paper. 

t  Evidence  connected  not  with  Goldbach's  theorem  itself  but  with  a  closely 
related  problem  concerning  pairs  of  primes  differing  by  2.     See  g  7. 

1  1-5985  is  Brun's  constant,  while  1-3203  is  2A. 


250  Mr  Hardy  and  Mr  Littlewood,  Note  on 

that  Vr  (n)  is  the  number  of  expressions  of  n  as  the  sum  of  r  primes 
Then  if  r  is  odd  we  have 

(5-11)  v,(n)  =  o()i>-') 

if  V  is  even,  and 

if  n  is  odd,  p  being  an  odd  prime  divisor  of  ??,  and 

(513)  B=n[i^^^^^^^. 

where  tn-  runs  through  all  odd  primes.     On  the  other  hand,  if  r  is 
even,  we  have 

(5-21)  Vr(n)  =  o{n''-^) 

if  n  is  odd,  and 

where 

(5-23)  c=n|i- 


(^-ir 

if  71  is  even.     The  last  formula  reduces  to  (1)  of  Shah  and  Wilson's 
paper  when  r  =  2. 

We  have  not  been  able  to  find  a  rigorous  proof,  independent 
of  all  unproved  hypotheses,  of  any  of  these  formulae.  But  we  are 
able  to  connect  them  in  a  most  interesting  manner  with  the  famous 
'  Riemann  hypothesis '  concerning  the  zeros  of  Riemann's  function 
f  (5).  The  Riemann  hypothesis  may  be  stated  as  follows  :  ^(s)  has 
no  zeros  whose  real  part  is  greater  than  ^.  If  this  be  so,  it  follows 
easily  that  all  the  zeros  of  ^(s),  other  than  the  trivial  zeros  s  =  —  2, 
s  =  — 4, ...,  lie  on  the  line  <t  =  'R{s)  =  ^.  It  is  natural  to  extend 
this  hypothesis  as  follows:  no  one  of  the  functions  defined,  luhen  a-  >  1, 
hy  the  series 

n" 

possesses  zeros  luhose  real  part  is  greater  than  ^.  We  may  call  this 
the  extended  Riemann  hypothesis.  This  being  so,  what  we  can  prove 
is  this,  that  if  the  extended  Riemann  hypothesis  is  true,  then  the 
formidae  (5"11) — (5'23)  a?'e  true  for  all  values  of  r  greater  than  4. 
The  reasons  for  supposing  the  extended  hypothesis  true  are 
of  the  same  nature  as  those  for  supposing  the  hypothesis  itself 
true.  It  should  be  observed,  however,  that  it  is  necessar}",  before 
we  generalise  the  hypothesis,  to  modify  the  form  in  which  it  is 
usually  stated;  for  it  is  not  proved  (as  it  is  for  ^{s)  itself)  that 
L{s)  can  have  no  real  zero  between  ^  and  1. 


1 


Messrs  Shall  and  Wilson's  paper  251 

6.  A  modification  of  our  method  enables  us  to  attack  a  closely 
related  problem,  that  of  the  existence  of  pairs  of  primes  differing 
by  a  constant  even  number  k. 

We  have 

2  A  (n)  A  (n  +  k)  r^''+''  =  J-  f '"  \f{re^^)  \ '  e'^'^  cW, 

where  f(x)  is  the  same  function  as  in  §  1,  and  r  is  positive  and  less 
than  unity.  We  divide  the  range  of  integration  into  a  number  of 
small  arcs,  correlated  in  an  appropriate  manner  with  a  certain 
number  of  the  points  e"P''^"J,  and  approximate  to  {/(j'e'")!^  on  each 
arc  by  means  of  the  formula  (2-8).  The  result  thus  suggested  is 
that 

^A{n)A  (n  +  k)  r-  ^  ^^~  U  (^  £  ^)  , 

where  A  has  the  same  meaning  as  in  §  2  and  p  is  an  odd  prime 
divisor  of  k.     From  this  it  would  follow  that 

(6-1)  S    A  (v)  A  (v  +  k)  -  2AnU  P^)  ; 

and  that,  if  A^^.  (?2)  is  the  number  of  prime  pairs  less  than  /?,  whose 
difference  is  k,  then 

T.r  /  N        2ylri    „  /p  —  1\ 

(6-2)  ^V'«~(i^^,n(P-2). 

This  formula  is  of  exactly  the  same  form  as  (1),  except  that  p  is 
now  a  factor  of  k  and  not  of  n.     In  particular  we  should  have 

,_  .  ,         2An 
(6-3)  ^^(»)~(lo-g»r 

and 

(6-4)  ^^"<»>~(Ttg-LV 

We  should  therefore  conclude  that  there  are  about  two  pairs  of 
primes  differing  by  6  to  every  pair  differing  by  2.  This  conclusion 
is  easily  verified.  In  fact  the  numbers  of  pairs  differing  by  2,  below 
the  limits* 

100,    500,    1000,    2000,    3000,   4000,   5000, 
are 

9,    24,   35,    61,   81,    103,    125; 

while  the  numbers  of  pairs  differing  by  6  are 

16,    47,    73,    125,    168,   201,    241. 

*  To  be  precise,  the  numbers  of  pairs  ( j),  p')  such  that  p'  =p  +  2  and  p'  does  not 
exceed  the  limit  in  question. 


252  My^  Hardy  and  Mr  Littlewood,  Note  on 

The  numbers  of  pairs  differing  by  4,  which  should  be  roughly  the 
same  as  those  of  pairs  differing  by  2,  are 

9,    26,    41,    63,    86,   107,    121. 


7, 


Brun,  ni  his  note  ah'eady  referred  to,  recognises  the  corre- 
spondence between  the  problem  of  §§  2—4  and  that  of  the  prime- 
pairs  differing  by  2,  and  realises  the  identity  of  the  constants  in- 
volved m  the  formulae  ;  but  does  not  allude  to  the  more  o-eneral 
problem  of  prime-pairs  differing  by  k.  He  does  not  determme  the 
fundamental  constant  A,  attempting  only  to  approximate  to  it 
empirically  by  means  of  a  count  of  prime-pairs  differing  by  2  and 
less  than  100000,  made  by  Glaisher  in  1878*.  The  value  of  the 
constant  thus  obtained  is,  as  was  pointed  out  in  §  4,  seriously  in 
error.  The  truth  is  that  when  we  pass  from  (6-1),  which,  when 
k  =  2,  takes  the  form 

2    A{v)A{v  +  2)r^^An, 

to  (6-3),  the  formula  which  presents  itself  most  naturally  is  not 
(6-3)  but  "^ 

(7-1)  i\r3(n)o.2yir--^. 

J    (log^)- 

This  formula  is  of  course,  in  the  long  run,  equivalent  to  (6-3) 
But 

(log  xf      (log  ny  \    ^  log  n  "^  (log  nf  "^  " '  7    ' 

and  the  second  factor  on  the  right-hand  side  is,  for  n  =  100000  far 
from  negligible.  Thus  (6-3)  may  be  expected,  for  such  values  of 
n,  to  give  results  considerably  too  small. 

}^C^^  *^^®  *^®  ^^^^®^^  ^"^^^*  ^^  integration  in  (7-1)  to  be  2  we 
find  that  the  value  of  the  right-hand  side  for  n  =  100000  is  to' the 
nearest  integer,  1249,  whereas  the  actual  value  of  i\^.,  (92)  is,  accord- 
ing to  Glaisher,  1224^  The  ratio  is  1-02,  and  the  agreement  seems 
to  be  as  good  as  can  reasonably  be  expected. 

Tir  "Sf  ^^l^^^ation  of  prime-pairs  has  been  carried  further  by 
Mrsfetreatteild,  whose  results  are  exhibited  m  the  following  table: 

*'ri        '^^■-  •     ^^e  number  of  pairs  below  100000  is  1225 

t  iiie  series  is  naturally  divergent,  and  must  be  closed,  after  a  finite  number  of 
terms   with  an  error  term  of  lower  order  than  the  last  term  retained 

^  Glaisher  reckons  1  as  a  prime  and  (1,  3)  as  a  prime-pair,  making  1225  in  all. 


Messrs  Shah  and  Wilsons  paper 


253 


11 

N,(n) 

2  A   f"  ''"■ 

Ratio  1 

1 

100,000 

1224 

1249 

! 

1-020 

200,000 

2159 

2180 

1-010 

300,000 

2992 

3035 

1-014 

400,000 

3801 

3846 

1-012 

500,000 

1562 

4625 

1-014 

600,000 

5328 

5381 

1-010 

8.  In  a  later  paper*  Brun  gives  a  more  general  formula  relating 
to  prime-pairs  {p,  p)  such  that  p  =  ap  +  2.  This  formula  also 
involves  an  undetermined  constant  k.  It  is  worth  pointing  out 
that  our  method  is  equally  applicable  to  this  and  to  still  more 
general  problems.  Suppose,  in  the  first  place,  that  v{n)  is  the 
number  of  expressions  of  n  in  the  form 

n  =  ap  +  hp, 

where  pi  and  p'  are  primesf.     We  may  suppose  without  loss  of 
generality  that  a  and  h  have  no  common  factor. 

The  results  suggested  by  our  method  are  as  follows.  If  n  has 
any  factor  in  common  with  a  and  h,  then 


"<">='' {(log. o-^}' 


and  this  is  true  even  when  n  is  prime  to  both  a  and  h,  unless  one 
of  n,  a,  b  is  even|.  But  if  n,  a  and  b  are  coprime,  and  one  of  them 
even,  then 


2J. 


n 


P-i 


ab  (log iif       \p—  2 

where  A  is  the  constant  of  §  2,  and  the  product  is  now  extended 
over  all  odd  primes  which  divide  n  or  a  or  b. 


*  '  Sur  les  nombres  premiers  de  la  forme  ap  +  h\  Archiv  for  Mathematik,  vol. 
24,  1917,  no.  14. 

t  We  might  naturally  include  powers  of  primes. 

+  These  results  are  trivial.  If  n  and  a  have  a  common  factor,  it  divides  hp', 
and  is  therefore  necessarily  p' ,  which  can  thus  assume  but  a  finite  number  of  values. 
If  n,  a,  h  are  all  odd,  either  ^^  ox  p'  must  necessarily  be  2. 


254  Mr  Hardy  and  Mr  Littleiuood,  Note  etc. 

Similarly,  suppose  N(n)  to  be  the  number  of  pairs  of  solutions 
of  the  equation 

a})'  —  hp  =  h 

such  that  p'  <  n.     It  is  supposed  that  a  and  h  have  no  common 
factor.     Then 

N(n)  =  o 


l(logw)- 

unless  k  is  prime  to  both  a  and  b,  and  one  of  the  three  is  even. 
If  these  conditions  are  satisfied 

where  p  is  now  an  odd  prime  factor  of  k,  a,  or  h. 


Mr  Harrison,  The  distribution  of  Electric  Force,  etc.       255 


The  distribution  of  Electric  Force  bettueen  two  Electrodes,  one  of 
which  is  covered  with  Radioactive  Matter.  By  W.  J.  Harrison, M.A., 
Fellow  of  Clare  College. 

[Read  17  February  1919.] 

It  has  been  shown  by  Rutherford*  that  it  is  probable  that  the 
ionisation  due  to  an  a  particle  per  unit  length  of  its  path  is  in- 
versely proportional  to  its  velocity,  provided  the  velocity  exceeds 
a  certain  minimum  necessary  to  effect  ionisation.  It  follows  that 
the  ionisation  per  unit  time  is  constant  at  all  points  of  the  path. 

Suppose  radioactive  matter  distributed  uniformly  over  the  sur- 
face of  a  large  plane  electrode  assumed  to  be  infinite  in  order  to  obtain 
simplicity  in  calculation.  Consider  the  a  particles  projected  from  a 
point  P  of  the  electrode.  These  particles  are  projected  equally  in  all 
directions,  hence  the  rate  of  ionisation  per  unit  volume  at  a  point 
Q  will  be  proportional  to  l/PQ^  provided  PQ<  R,  where  R  is  the 
range  of  the  particles.  The  total  rate  of  ionisation  at  a  point  Q 
distance  x  (j:  <  R)  from  the  electrode  will  be  proportional  to 

■\/Ji---'e"27rrdr 


0  x-  +  r^' 

where  r  is  the  distance  of  a  point  P  on  the  electrode  from  the  foot 
of  the  perpendicular  from  Q.     Now 

•sjitr—x^  27' dr 


Jo 


x^  +  r'' 

=  log 


log  (x-  +  r^) 
R 


sJlP-X' 

0 


Hence  rate  of  ionisation 

1      ^ 

^  =  ^0  log  -;  . 


X 


The  equations  determining  the  distribution  of  electric  force  are 
given  by  Thomson,  Conduction  of  Electricity  through  Gases,  1906, 
chap.  III.  The  notation  of  this  book  is  adopted  as  being  sufficiently 
well  known.  The  differential  equation  for  the  electric  force  X  is  of 
the  form 

d'X^^       a  (dX-'\'       b  ,      R  „ 

_—  =0,  x>  R. 

dx~ 

*  Radioactive  Substances  and  their  Radiations,  1913,  p.  158. 
VOL.  XIX.  PART  v.  18 


256    Mr  Harrison,  The  distribution  of  Electric  Force  hetiueen  ttuo 

The  numerical  solution  may  be  obtained  for  any  particular 
values  of  the  constants  a,  b,  c,  q^,  R  by  approximate  methods.  In 
the  absence  of  any  definite  experimental  results  with  which  to 
compare  the  calculations,  the  labour  involved  in  integration  is  not 
worth  undertaking. 

The  case,  however,  of  the  saturation  current  is  the  most  impor- 
tant, and  the  integration  is  simple.  It  is  assumed  that  recombi- 
nation of  ions  does  not  take  place  in  this  case,  and  therefore  the 
equations  reduce  to 

=  0,  x>R. 

Write  Si7eq,{^  +  y\=K. 

Then,  for  x<  R, 


for  X  >  R, 


9  log  7.  -\-ix-  +  Bx  +  G 


(vide  Rutherford,  Radioactive  Substances,  etc.,  p.  67),  A,  B,  G  are 
constants  of  integration. 
Now  the  conditions  are 

(1)  at  ^  =  0,  ni  =  0,  if  ^  =  0  be  the  positive  plate, 

(2)  at  a;  =  J?,  Wg  =  0, 

(3)  at  a;  =  i^,  n^  is  continuous, 

(4)  Sit  x  =  R,  X  \B  continuous. 

{vide  Conduction  of  Electricity  through  Gases,  chap.  iii.). 

dX^  _  _  Sttj 
'  '     dx  kz  ' 

(2)  and  (3)  lead  to  the  same  condition,  which  is  the  same  as 
(1),  if 

i  =  eRqQ. 

Now  since  there  is  no  recombination 


.      [^  R 

1=1    eqolog  -  dx  =  eRqo. 
Jo  ^ 


Electrodes,  one  of  which  is  covered  luith  Radioactive  Matter  257 


Hence  conditions  (1),  (2),  (3)  are  identical  and  determine  B. 
Condition  (4)  supplies  a  relation  between  G  and  A, 

A^K{G-IR^). 
Hence 


X'  =  K 


^x~  log h  f 


h 


Rx  +  BR' 


X'  =  K 


CO  R. 


X  ft-l  ~f~  rCo 

0<x<R,  where  BR-  =  C, 

The  constant  B  can  be  determined  when  the  potential  differ- 
ence between  the  electrodes  is  given*. 

The  general  character  of  these  results  can  be  shown  by  numerical 
calculation  for  the  cases  k,  =  L,  l-25k,=k^,  A;i  =  1-25 A-.,  (corresponding 
to  the  case  in  which  the  positive  ion  moves  more  slowly,  as  usual, 
than  the  negative  ion,  and  the  radioactive  matter  is  spread  on  the 
negative  plate),  and  for  distances  R,  2R,  SR  between  the  electrodes, 
and  for  B  =  O'l,  0-5,  I'O.  In  order  that  the  current  may  be  the  satu- 
ration current  it  is  necessary  in  practice  that  B  should  exceed 
a  certain  limit.  This  limit  is  dependent  on  the  particular  conditions 
of  any  given  experiment. 

The  distribution  of  the  electric  force  X  is  shown  on  the  graph 
below.     The  curves  marked  (1),  (2),  (3)  are  for  the  cases 
kj  =  1-25  k.2,  ki  =  ^'2,  A.-2  =  1-25  k^,  respectively. 

The  potential  difference  V  between  the  electrodes  is  given  in 
the  following  table,  d  being  the  distance  between  the  plates. 

V 
RKK^' 


ki  =  l-25k.2 

^1  =  ^2 

k.2^1-25ki 

D=0-l 

d=R 

d=2R 

d=3R 

-343 

1-056 
2-034 

-379 
1-147 
2-193 

•412 
1-232 
2-343 

D  =  Ob 

d=R 

d=2R 

d=3R 

-725 
1-676 
2-841 

-743 

1-740 
2-963 

-762 
1-800 
3-078 

D  =  l-0 

d=R 

d=2R 

d=3R 

1-014 
2-203 
3-566 

1-027 
2-250 
3-663 

1-041 
2-298 
3-759 

These  forms  of  X  are  not  strictly  valid  in  the  immediate  neighbourhood  of  the 
electrodes,  as  the  natural  agitation  of  the  ions  has  been  neglected  in  this  theory 
Vide  Pidduck,  Treatise  on  Electricity,  1916,  p.  505. 

18—2 


258       Mr  Harrison,  The  distribution  of  Electric  Force,  etc. 


I  '0  R  Z'O  R 

DISTANCE     BETWEEN     ELECTRODE^. 


3-0  R 


Mr  Purvis,  The  conversion  of  saiv-dust  into  sugar         259 


The  conversion  of  saw-dust  into  sugar.    By  J.  E.  PuRViS,  M.A. 
[Read  17  February  1919.] 

The  production  of  sugar  from  wood  is  well  known.  In  the 
Classen  process,  saw-dust  is  digested  in  closed  retorts  with  a  weak 
solution  of  sulphurous  acid  under  a  pressure  of  between  six  and 
seven  atmospheres.  The  products  contain  about  25  °/^  of  dextrose, 
and  other  substances  are  pentose,  acetic  acid,  furfurol  and  formal- 
dehyde. Cellulose  material  can  also  be  converted  into  sugar  by 
other  acids. 

The  following  results  were  obtained  by  digesting  saw-dust 
from  ordinary  deal  with  different  acids  of  varying  concentrations ; 
estimating  the  amount  of  sugar  in  the  liquid  in  the  usual  way 
from  the  amount  of  cuprous  oxide  precipitated  from  Fehling's 
solution,  and  converting  this  oxide  of  copper  to  cupric  oxide.  The 
numbers  were  then  calculated  in  terms  of  dextrose. 

(1)  25  grams  of  saw-dust  were  digested  with  300  c.c.  distilled 
water  and  50  c.c.  strong  H2SO4  (1  c.c.  H2S04  =  1*78  grms.  H2SO4) 
for  5^  hours  in  a  sand  bath  at  a  temperature  just  below  the 
boiling  point  and  the  mixture  was  constantly  stirred.  This  was 
then  filtered ;  the  residue  well  washed  and  the  filtrate  made  up  to 
a  litre ;  10  c.c.  of  the  filtrate  were  neutralised  with  sodium 
carbonate  and  the  cuprous  oxide  from  Fehling's  solution  was 
precipitated,  filtered,  dried  and  ignited  to  cupric  oxide.  This  gave 
0"215  grm.  CuO  which  is  equivalent  to  39  °/^  of  dextrose  on  the 
original  amount  of  saw-dust. 

(2)  25  grams  of  saw-dust  to  which  were  added  500  c.c.  of 
distilled  water  and  25  c.c.  of  strong  H2SO4  of  the  same  strength  as 
in  experiment  (1)  and  digested  for  5  hours  under  the  same 
conditions.    This  gave  13  °/^  of  dextrose. 

(3)  50  grams  of  saw-dust  were  digested  with  500  c.c.  of 
distilled  water  and  50  c.c.  of  the  strong  H2SO4  for  5f  hours.  The 
yield  was  11  "5  %  dextrose. 

(4)  25  grams  of  saw-dust  were  digested  with  250  c.c.  of  tap 
water  and  10  c.c.  of  strong  H2SO4  for  2  hours.  This  yielded  10*5  7o 
dextrose. 

(5)  25  grams  of  saw-dust  were  digested  with  720  c.c.  of  tap 
water  and  10  c.c.  strong  H2SO4  for  2  hours.  This  produced  3*35  "/^ 
dextrose. 

(6)  50  grams  of  saw-dust  were  digested  with  500  c.c.  water 
and  50  c.c.  N/1  HCl  (=  1-825  grms.  HCl)  for  3  hours.  This  gave 
3-35  %  dextrose. 


260  Mr  Purvis,  The  conversion  of  saw-dust  into  sugar 

(7)  50  grams  of  saw-dust  were  digested  with  500  c.c.  water 
and  100  c.c.  N/1  H2SO4  (=  2-45  grms.  H.SO4)  for  2  hours.  This 
produced  1'82  °/^  dextrose. 

(8)  25  grams  of  saw-dust  were  digested  with  700  c.c.  water 
and  5  grams  P0O5  for  12  hours  at  the  temperature  of  the  room 
(about  15°  C),  and  then  for  3  hours  just  below  the  boiling  point. 
This  gave  12'66  °/^  dextrose. 

The  results  show  that  the  amount  of  sugar  which  can  be 
obtained  depends  on  the  nature  of  the  acid  and  its  strength  relative 
to  the  amount  of  saw-dust,  and  on  the  time  of  digestion.  The 
greatest  amount  was  obtained  when  the  strongest  sulphuric  acid 
acted  for  a  considerable  time.  In  the  other  experiments  not  so 
much  was  obtained  as  by  the  Classen  process.  For  the  commercial 
production  of  sugar  from  such  a  cheap  material  as  saw-dust  the 
question  to  be  decided  would  be  the  relative  cost  of  the  Classen 
process  compared  with  the  cost  under  the  conditions  of  these 
experiments.  That  would  include  a  comparison  of  the  cost  of 
the  various  acids  and  the  recovery  of  these  acids  for  further  use. 
The  conversion  of  sugar  into  alcohol  and  acetone  presents  no 
difficulty ;  and  it  would  be  important  to  consider  whether  such 
useful  chemical  substances  could  not  be  produced  from  a  waste 
product  like  saw-dust  at  a  cheaper  rate  than  by  the  present  costly 
methods. 


Mr  Purvis,  Bracken  as  a  source  of  potash  261 


Bracken  as  a  source  of  potash.     By  J.  E.  Purvis,  M.A. 
[Read  17  February  1919.] 

The  Master  of  Christ's  College,  Cambridge,  in  the  autumn  of 
1917,  had  some  correspondence  with  Mr  J.  A.  A.  Williams  of 
Aberglaslyn  Hall,  Beddgelert,  in  regard  to  the  use  of  bracken  as  a 
fertiliser.  Mr  Williams  had  burnt  the  bracken  growing  on  a  peaty 
soil  on  his  estate  at  Beddgelert,  ploughed  in  the  ashes  and  obtained 
highly  satisfactory  crops  of  potatoes.  It  seemed  to  be  of  some 
importance  to  find  out  what  amount  of  potash  could  be  obtained 
from  the  ash;  and  in  October  1917  a  sample  of  bracken  from  the 
Botanic  Gardens,  Cambridge,  was  analysed.  This  grows  on  a  poor 
sandy  soil. 

It  is  known  that  bracken  contains  larger  quantities  of  potash 
in  the  summer  months  than  in  the  autumn  and  more  complete 
investigations  were  deferred  till  the  summer  of  1918.  Meanwhile 
in  the  April  (1918)  number  of  the  Journal  of  Agriculture  (vol.  25, 
no.  1,  p.  1)  Messrs  Berry,  Robinson  and  Russell  published  an 
article  on  "  Bracken  as  a  source  of  potash  "  which  contained  the 
results  of  the  analyses  of  material  collected  from  various  districts 
in  England,  Scotland  and  Wales  from  May  to  October  1916,  and 
from  June  to  October  1917.  The  numbers  show  that  the  amount 
of  potash  is  much  higher  in  the  summer  months  than  in  the  autumn. 
For  example,  bracken  gathered  June  1st,  1917,  from  Harpenden 
Common,  Rotharnsted,  which  is  mainly  gravel  and  clay,  produced 
4"1  ° I ^  of  potash  (KoO)  on  the  dried  material  and  only  1'8  7o  when 
gathered  September  1st,  1917.  The  authors  also  considered  that 
their  evidence  indicates  a  more  rapid  falling  off  of  the  potash  from 
bracken  growing  on  sandy  and  peaty  soils  than  on  heavier  soils 
rich  in  potash  :  and  that,  therefore,  its  chances  of  success  as  a 
fertiliser  would  be  greater  in  these  heavier  soils. 

In  view  of  these  results  the  investigations  were  continued  with 
the  bracken  growing  in  the  Botanic  Gardens,  Cambridge,  and  also 
with  that  on  Mr  Williams's  Welsh  estate.  The  following  tables 
summarise  the  results. 

Generally,  the  numbers  are  of  the  same  order  as  those  obtained 
by  Messrs  Berry,  Robinson  and  Russell,  and  confirm  the  opinion 
that  in  the  summer  months  there  is  more  potash  than  in  the  later 
months.  Also  there  is  a  clear  indication  that,  on  an  average,  the 
Welsh  peaty  soil  yields  more  potash  than  the  Cambridge  poor 
sandy  soil. 


262  Mr  Purvis,  Bracken  as  a  source  of  potash 

Cambridge  Bracken. 


Date  when  sample 
was  gathered 

Percentage  of 
dry  matter  in 
fresh  bracken 

Percentage  of 

ash  in 

dry  matter 

Percentage  of  potash  (K2O)  in 

fresh  bracken 

dry  bracken 

16  October,  1917 

27-60 

7-51 

0-29 

0-82 

1  June,  1918 

15-34 

6-81 

0-46 

3-00 

2  July,  1918 

21-58 

5-02 

0-52 

2-45 

1  August,  1918 

30-26 

5-96 

0-50 

1-70 

31  August,  1918 

26-50 

7-86 

0-30 

1-07 

1  October,  1918 

29-06 

7-93 

0-33 

1-13 

Welsh  Bracken 

3  June,  1918 

24-4 

6-55 

0-77 

3-19 

4  July,  1918 

25-8 

5-78 

0-83 

3-22 

31  July,  1918 

40-7 

3-84 

0-42 

1-45 

1  September,  1918 

30-97 

7-02 

0-53 

1-71 

3  October,  1918 

34-54 

4-82 

0-45 

1-32 

To  estimate  the  cost  of  collection  is  difficult  as  the  conditions 
of  transit  and  labour  are  variable  and  estimates  for  one  locality 
would  be  useless  for  another.  It  is  evident,  however,  that  bracken 
is  a  valuable  source  of  potash :  but  its  economic  application  as  a 
fertiliser  will  be  controlled  by  the  requirements  and  conditions  of 
the  neighbourhood  where  it  grows. 

I  have  to  thank  Mr  Williams  for  supplying  the  Welsh  bracken, 
and  Mr  Lynch,  of  the  Cambridge  Botanic  Gardens,  for  samples 
from  the  gardens. 


Dr  Shearer,  The  action  of  electrolytes  on  the  electrical,  etc.  263 


The  action  of  electrolytes  on  the  electrical  conductivity  of  the 
bacterial  cell  and  their  effect  on  the  rate  of  migration  of  these  cells 
in  an  electric  field.  By  C.  Shearer,  Sc.D.,  F.R.S.,  Clare  College. 
(From  the  Pathological  Laboratory,  Cambridge.) 

{Read  17  February  1919.] 

If  a  thick  creamy  emulsion  of  the  meningococcus  or  B.  coli  is 
made  up  in  neutral  Ringer's  solution  (that  is,  one  in  which  the 
sodium  bicarbonate  is  left  out),  and  the  conductivity  measured  by 
means  of  a  Kohlrausch  bridge  and  cell;  it  is  found  that  its  resistance 
is  more  than  treble  that  of  the  same  solution  without  the  bacteria : 
that  is  the  greater  part  of  the  resistance  is  due  to  the  presence  of 
the  bacteria. 

This  determination  was  made  as  follows:  a  24  hour  culture  of 
the  meningococcus  or  B.  coli  on  trypagar  (2-t  plates)  was  washed 
off  in  a  considerable  quantity  of  Ringer's  solution,  centrifuged  down 
and  re  washed  several  times  in  a  similar  manner  to  remove  all  traces 
of  serum  or  any  salts  derived  from  the  culture  medium.  The  centri- 
fuged deposit  was  then  made  up  to  standard  strength  in  neutral 
Ringer's  solution,  so  that  it  was  not  too  thick  to  be  sucked  up  in  a 
medium  sized  pipette  and  transferred  to  a  Hamburger  cell  and  its 
conductivity  determined.  It  was  found  that  the  conductivity  of 
such  standard  emulsions  when  measured  under  similar  conditions 
of  temperature  was  fairly  uniform*.  When  sufficient  care  was 
taken  to  get  the  emulsions  of  the  right  thickness,  resistances  of 
110  ohms  could  be  pretty  constantly  obtained.  The  same  quantity 
of  Ringer's  solution  alone  had  about  26"7  ohms  resistance  under 
the  same  conditions. 

If,  however,  in  place  of  the  Ringer's  solution  we  make  up  the 
bacterial  emulsions  in  pure  sodium  chloride  of  the  same  conducti- 
vity as  that  of  the  Ringer's  solution,  i.e.  one  in  which  the  resistance 
is  26'7  ohms  (which  corresponds  to  a  NaCl  solution  of  about  0'85  °/^), 
we  obtain  as  in  the  case  of  the  emulsion  in  Ringer's  solution  an 
initial  resistance  of  110  ohms.  Within  a  few  minutes,  however,  this 
gradually  drops  and  at  the  end  of  30  or  40  minutes  the  emulsion 
now  has  the  same  conductivity  as  that  of  the  bare  sodium  chloride 
solution  without  the  bacteria,  i.e.  26'7  ohms  resistance.  Thus  pure 
sodium  chloride  of  about  the  concentration  as  that  present  in  the 
blood  gradually  destroys  the  resistance  of  the  bacterial  cell.  If  the 
bacteria  are  allowed  to  lie  in  this  solution  for  several  hours  it  will 
be  found  that  at  the  end  of  this  time,  on  subculture,  they  are 

••"  All  measurements  were  made  at  constant  temperature  25°  C.   Resistance  con- 
stant of  conductivity  cell  =  29  8  x  10~^. 


264  Dr  Shearer,  The  action  of  electrolytes  on  the 

dead.  If  they  are  only  allowed  to  remain  in  the  NaCl  for  a  short 
time  and  then  transferred  to  neutral  Ringer  again  they  immediately 
return  to  their  normal  resistance  and  grow  freely  on  subculture. 

If  when  the  resistance  of  the  bacterial  emulsion  has  fallen  in 
NaCl  solution  a  little  CaCL  is  added  it  again  regains  its  normal 
conductivity  and  is  uninjured.  Thus  we  get  the  usual  antagonistic 
action  of  CaCla  to  NaCl.  It  was  found  that  KCl,  LiCl,  MgCl^ 
acted  like  NaCl  in  reducing  the  resistance  offered  by  the  bacteria, 
while  BaCls,  SrClg  have  no  action  on  the  resistance  but  act  like 
CaClg.  Thus  it  is  clear  that  in  the  bacteria  as  with  so  many  other 
plant  and  animal  cells  the  entrance  of  the  ions  of  NaCl,  KCl, 
LiCl,  MgCla  is  prevented  by  the  presence  of  very  small  quantities 
of  CaCL,  BaCla  or  SrClg.  Bacterial  emulsions  made  up  in  BaClo, 
SrCla  and  CaClg ,  having  the  same  conductivity  as  Ringer's  solution, 
showed  no  change  in  resistance  on  being  kept  in  these  solutions 
for  some  time,  invariably  remaining  normal. 

The  interest  of  these  experiments  consists  in  that  they  agree 
completely  with  the  results  obtained  by  Loeb,  Osterhout  and  a 
large  number  of  other  workers  on  animal  and  plant  cells. 

In  Laminaria,  Osterhout  finds  with  CaCL  and  presumably  also 
with  BaCla  and  SrCl..  there  is  invariably  a  brief  temporary  rise  in 
resistance  when  placed  in  these  solutions  of  the  same  conductivity 
as  sea-water  which  is  followed  by  a  gradual  fall.  With  the  bacterial 
cell  no  such  preliminary  rise  can  be  distinguished,  w^hile  the  fall 
due  to  the  toxic  action  of  the  solution  is  much  delayed  and  slower. 

In  view  of  the  remarkable  action  of  tri-valent  ions  on  artificial 
membranes  as  shown  by  the  work  of  Perrin,  Girard  and  Mines,  and 
the  action  on  the  permeability  of  cell  wall  as  shown  by  the  work 
of  Mines,  Osterhout  and  Gray,  it  is  of  great  interest  to  consider 
their  action  on  the  bacterial  cell. 

While  the  tri-valent  positive  ion  of  lanthanium  nitrate  brings 
about  a  rapid  rise  of  resistance  in  Laminaria  according  to  Osterhout 
and  in  the  Echinoderm  egg  according  to  Gray,  when  this  salt  is 
used  in  such  dilution  as  not  to  affect  the  conductivity  of  the  solu- 
tion itself,  no  such  action  can  be  distinguished  in  the  case  of 
bacteria  by  means  of  the  Kohlrausch  method.  The  resistance 
remains  unchanged  until  it  begins  to  fall  on  account  of  the  in- 
creasing strength  of  the  salt  added.  In  the  same  way  the  positive 
tri-valent  ions  of  CeCL,  neo-ytterbium  chloride  and  the  tri-valent 
negative  ions  of  sodium  citrate  appear  to  have  no  action  in  in- 
creasing or  decreasing  the  resistance  of  the  bacterial  cell  as  deter- 
mined by  the  conductivity  method.  It  should  be  pointed  out  that 
these  salts  can  only  be  used  in  very  dilute  solutions.  In  the  case 
of  lanthanium  nitrate  this  salt  readily  flocculates  living  bacteria 
Avhen  used  in  stronger  solutions  than  ^  y\j^  M. 

It  would  seem  remarkable  in  view  of  the  sharp  action  of  La  on 


electrical  conductivity  of  the  bacterial  cell,  etc.  265 

the  Echinoderm  egg  when  used  in  a  strength  of  g^*^  M.  that  some 
similar  action  should  not  be  found  with  bacteria,  but  repeated 
experiments  with  centrifuged  solid  bacterial  deposits  of  both  the 
meningococcus  and  B.  coli  using  the  same  type  of  electrodes  used 
by  Gray  for  the  Echinoderm  egg  and  obtaining  resistances  as  high 
as  150  ohms  failed  to  show  any  initial  rise  of  resistance.  It  was 
possible  that  in  the  case  of  bacteria,  their  enormous  surface  would 
render  the  preliminary  rise  of  resistance  so  temporary  that,  before 
the  electrodes  could  be  placed  in  position  and  the  bridge  readings 
adjusted,  it  would  be  over  and  passed.  To  test  this  point  a  small 
quantity  of  La  was  added  while  the  bridge  telephone  was  kept  to 
the  ear,  but  in  every  instance  no  change  could  be  detected.  It 
would  seem  that  the  bacterial  cell  is  normally  in  a  state  of 
maximum  impermeability  and  that  this  can  not  be  further  increased 
by  the  presence  of  CaCL  and  the  tri-valent  salts. 

In  distinction  to  the  absence  of  effect  of  the  tri-valent  salts  on 
bacteria  as  demonstrated  by  the  conductivity  method,  is  the  marked 
action  of  these  salts  and  especially  lanthanium  nitrate  in  changing 
the  rate  of  migration  of  these  cells  in  an  electric  field.  This  can 
be  determined  by  the  ultramicroscopic  or  still  better  the  U  tube 
method. 

If  10  c.c.  of  a  thick  growth  of  B.  coli  in  spleen  broth  be  run 
into  a  U  tube  under  neutral  Ringer's  solution  of  the  same  conducti- 
vity as  the  broth,  then  on  passing  an  electric  current  through  the 
tube,  the  temperature  being  constant,  an  even  rapid  migration  of 
the  bacteria  takes  place  towards  the  anode. 

That  practically  all  bacteria  carry  a  negative  charge  and  migrate 
to  the  anode  has  been  repeatedly  confirmed  by  numerous  workers, 
but  what  is  of  interest  here  is  that  this  charge  can  be  materially 
modified  by  various  tri-valent  salts,  especially  La.  If  to  the  10  c.c. 
of  B.  coli  emulsion  in  spleen  broth  run  into  the  U  tube  in  the 
above  experiment  1  c.c.  of  a  -^^  M.  lanthanium  nitrate  solution 
is  added,  it  will  be  found  that  the  rate  of  migration  of  the 
bacilli  under  the  same  conditions  of  electric  field  and  temperature 
is  now  halved.  If  2  c.c.  of  the  solution  is  added,  little  or  no  migra- 
tion takes  place  and  the  emulsion  soon  flocculates  and  is  preci- 
pitated to  the  bottom  of  the  tube. 

In  terms  of  the  Helmholtz-Lamb  theory  of  the  double  electric 
layer  the  addition  of  the  La  has  considerably  altered  the  nature  of 
the  charge  on  the  bacterial  cell  wall.  The  conductivity  method 
however  fails  to  show  any  change  under  this  condition.  This  result 
is  possibly  of  some  interest  in  view  of  Mines'  theory  of  the  polarising 
action  of  certain  ions  on  the  cell  membrane.  It  is  of  course  possible 
that  the  resistances  obtained  in  the  conductivity  experiments  were 
too  low  to  bring  out  the  real  changes  taking  place. 


266  Miss  Haviland,  The  bionomics  of  Aphis 


The  bionomics  of  Aphis  grossulariae  Kalt.,  and  Aphis  viburui 
Schr.  By  Maud  D.  Haviland,  Bathurst  Student  of  Newnham 
College.     (Communicated  by  Mr  H.  H.  Brindley.) 

[Read  17  February  1919.] 

Aphis  grossulariae  Kalt.  is  a  serious  pest  'of  currant  and  goose- 
berry bushes  in  this  country.  It  attacks  the  young  shoots  in  May, 
and  when  present  in  numbers,  it  distorts  them  to  such  an  extent 
that  growth  ceases  and  a  dense  cluster  of  leaves  is  formed,  under 
which  the  aphides  swarm. 

The  bionomics  of  this  aphis  are  incompletely  known.  It  appears 
on  red  currants  in  May,  and  remains  there  until  the  middle  or  end 
of  July.  The  sexuales  have  never  been  found.  In  1912  Theobald 
(Journ.  Econ.  Biol.,  vol.  Vli.  p.  100)  first  pointed  out  its  resemblance 
to  Aphis  viburni  Schr.,  a  common  species,  which  is  found  on  the 
guelder  rose  (  Viburnum  opulus)  in  spring  and  summer,  while  the 
sexual  forms  have  been  recorded  from  the  same  plant  in  the  autumn. 
Aphis  viburni  has  a  very  characteristic  appearance,  owing  to  the 
row  of  lateral  tubercles  on  the  abdomen.  Such  tubercles  are  not 
very  common  among  the  Aphidinae,  but  they  are  prominent  like- 
wise in  Aphis  grossulariae.  In  fact  there  seems  to  be  no  structural 
difference  between  the  two  species;  though  in  spirit  specimens,  the 
guelder  rose  aphis  frequently  stains  the  alcohol  dark  brown,  while 
the  currant  form  has  no  such  property. 

In  May  1918,  I  had  under  observation  some  red  and  black 
currant  bushes,  and  two  guelder  rose  shrubs,  which  all  grew  close 
together.  Early  in  the  month  all  were  free  from  aphid  attack,  but 
on  May  31st  three  colonies,  each  consisting  of  a  single  winged 
female  with  a  few  new-born  young,  appeared  on  the  guelder  roses, 
and  the  same  evening  four  sprigs  of  currant  were  likewise  each 
infected.  During  the  following  week,  numerous  other  winged  forms 
appeared  both  on  the  guelder  roses  and  on  the  currants.  The 
method  of  attack  was  the  same  in  both  cases.  The  migrant  crept 
into  the  axil  of  a  leaf,  and  from  thence  her  progeny  gradually  spread 
up  the  stem  and  along  the  midrib.  About  the  same  time,  I  found 
a  Viburnum  tree  swarming  with  winged  females  of  Aphis  viburni 
in  a  shrubbery  a  hundred  yards  away;  and  as  these  were  in- 
distinguishable from  the  migrants  on  the  Viburnum  and  currants, 
I  have  little  doubt  that  this  was  the  source  of  infection. 

Assuming  that  A.  viburni  and  A.  grossulariae  are  identical,  I 
began  experiments  to  test  how  far  the  host  plants  were  interchange- 
able. Unfortunately,  owing  to  heavy  rains,  the  experiments  with 
the  original  winged   migrants  were  all  inconclusive,  and  during 


grossulariae  Kalt.,  and  Aphis  viburni  Schr.  267 

June  and  July  I  worked  with  alate  and  apterous  individuals  of  later 
generations.  The  results  are  set  out  in  the  accompanying  tables 
from  which  it  will  be  seen  that  out  of  thirteen  attempts  to  transfer 
A.  viburni  to  Ribes  rubrum,  in  only  two  cases  did  the  resulting 
colonies  survive  more  than  ten  days,  while  reproduction  was  very 
feeble  and  never  occurred  beyond  the  third  generation.  In  one 
case  (Table  A,  Number  IX)  an  attempt  was  made  to  re-transfer  the 
third  generation  back  from  the  currant  to  the  guelder  rose,  but 
the  result  was  that  the  aphides  all  died  within  twenty-four  hours. 

Similar  attempts  were  made  to  transfer  A.  grossulariae  from 
currant  to  guelder  rose,  but  the  colonies  never  survived  more  than 
six  days,  and  reproduction  was  very  feeble.  Meanwhile  the  natural 
colonies  on  guelder  rose  and  currant  flourished  from  the  end  of 
May  to  the  middle  of  August  and  end  of  July  respectively. 

Aphis  grossulariae  has  not  been  recorded  from  other  food  plants, 
but  during  June  I  observed  three  instances  where  winged  migrants 
had  established  themselves  on  the  flower  heads  of  the  Canterbury 
Bell  {Campanula)  and  the  resulting  colonies  persisted  for  two  or 
three  weeks. 

The  conclusions  suggested  by  the  foregoing  observations  are 
that,  as  Theobald  points  out,  A.  grossulariae  is  probably  identical 
with  A.  viburni.  The  first  migrant  from  the  birth  plant  (  Viburntmi) 
can  form  colonies  either  on  Viburmim,  which  is  the  natural  host, 
or  else  on  Ribes.  The  descendants  of  the  migrants  to  Viburnum 
may  with  some  difficulty  be  established  on  currant  although  the 
resulting  colonies  are  not  so  strong  as  those  derived  from  an  early 
migrant.  On  the  other  hand  the  descendants  of  the  migrants  to 
currant  cannot  be  re-established  on  Viburnum.  It  seems  as  if  in 
two  or  three  generations  some  change  takes  place  in  the  currant 
form  which  prevents  it  from  flourishing  on  the  guelder  rose.  One 
explanation  is  that  there  is  some  change  in  the  constitution  of  the 
guelder  rose  plant — an  increase  of  tannins  for  instance — and  that 
the  strain  on  guelder  rose  can  gradually  adapt  itself  to  altered 
conditions  which  the  newly  transferred  currant  reared  stock  cannot 
tolerate.  But  this  explanation  is  not  wholly  satisfactory  because 
the  dates  show  that  unsuccessful  transferences  took  place  in  the 
second  and  third  generations  while  the  plants  were  still  young, 
while  the  most  successful  attempt  was  made  in  July  when  the 
shoots  were  mature.  It  is  also  worth  noticing  that  while  the  more 
successful  attempts  were  made  with  winged  parents,  yet  in  several 
of  the  Viburnum-io-cnvYajxt  experiments,  wingless  females  were 
found  to  feed  and  reproduce  on  the  new  host. 

Theobald  {op.  cit.  p.  100)  suggests  that  A.  grossulariae  maybe 
the  alternating  form  of  A.  viburni,  but  says  that  he  has  twice 
failed  to  transfer  the  former  to  Vibui^num — a  result  confirming  my 
own  experiments  in  Table  B.    On  the  other  hand,  it  is  possible  that 


Table  A. 

Results  of  transference  of  Aphis  viburni  from  Viburnum 

opulus  to  Ribes  rubrum. 


Number 

Date  of 
transference 

Forms  transferred 

Death  of 
last  survivor 

Number  of 

Generations  born 

On  new  host 

I 

12  .  VI .  18 

alate  and  apterous 

21  .  VI . 18 

?2 

II 

13  .  VI .  18 

alate 

17.  VI. 18 

1 

III 

17  .  VI . 18 

alate  and  apterous 

22  .  VI . 18 

?2 

IV 

24.  VI. 18 

apterous 

29  .  VI .  18 

1 

V 

29  .  VI . 18 

apterous 

2  .  VII .  18 

0 

VI 

13.  VI. 18 

alate  and  apterous 

26  .  VI . 18 

?3 

VII 

5  .  VII .  18 

apterous 

6.  VII.  18 

1 

VIII 

30  .  VI .  18 

— 

9  .  vii .  18 

2 

IX 

9  .  VII .  18 

alate  and  apterous 

25.  VII.  18 

3 

X 

6  .  VII .  18 

apterous 

12.  VII.  18 

2 

XI 

5  .  VII .  18 

apterous 

6  .  VII .  18 

0 

XII 

6  .  VII .  18 

— 

7  .  VII .  18 

0 

XIII 

9.  VII.  18 

13.  VII.  18 

?1 

Table  B. 

Table  of  transference  of  Aphis  viburni,  self-established  on 

Ribes  rubrum,  to  Viburnum. 


Number 

Date  of 
transference 

Forms  transferred 

Death  of 

last  survivor 

Number  of 

Generations  born 

on  new  host 

I 

5  .  VI .  18 

apterous 

12. VI. 18 

1 

II 

2  .  VI . 18 

alate  and  apterous 

8  .  VI . 18 

1 

III 

8  .  VI . 18 

— 

10  .  VI . 18 

— 

IV 

10  .  VI . 18 

— 

14.  VI. 18 

1 

V 

22  .  VI . 18 

apterous 

24  .  VI . 18 

— 

VI 

30  .  VI .  18 

alate  and  apterous 

1  .  VII .  18 

— 

VII 

1  .  VII .  18 

apterous 

2.  VII.  18 

— 

VIII 

24  .  VII .  18 

alate  and  apterous 

25.  VII.  18 

— 

Miss  Haviland,  The  bionomics  of  Aphis  grossulariae,  etc.      269 

A.  grossulariae  is  not  the  natural  summer  form  of  A.  viburni,  but 
is  merely  a  casual  parasite  of  the  currant.  In  those  of  the  Aphidinae 
which  have  a  regular  migration  between  two  plants,  the  change  is 
usually  from  a  woody  stemmed  primary,  to  a  herbaceous  secondary, 
host;  and  if  in  the  case  of  ^.  viburni,  the  currant  should  be  found 
to  be  the  normal  second  host,  it  would  be  a  remarkable  exception  to 
this  rule.  Perhaps  we  have  here  a  form  that  has  not  yet  adapted 
itself  to  the  conditions  of  modern  fruit  growing.  In  a  natural  state, 
the  aphides  are  probably  able  to  follow  the  whole  life  cycle  on 
Viburnum,  but  the  spread  of  the  cultivated  currant  has  presented 
them  with  an  increasing  supply  of  alternative  food  which  induces 
a  change  that  makes  a  return  to  Viburnum  impossible.  Whether 
sex-producing  forms  can  arise  from  the  currant  stock,  and  thence 
return  to  the  guelder  rose,  is  not  known.  If  not,  and  the  early  date 
of  the  disappearance  from  the  currant  is  against  this  view,  we  must 
consider  that  the  infestation  of  the  currant  is  an  unfortunate 
accident  in  the  history  of  the  species,  which  entails  a  waste  of 
migrating  individuals  upon  a  cultivated  plant  that  might  otherwise 
have  perpetuated  themselves  on  the  natural  host.  However  this 
does  not  mitigate  the  danger  of  the  pest  from  a  fruit  grower's  point 
of  view,  and  infected  Viburnum  ought  not  to  be  allowed  in  the 
neighbourhood  of  currant  bushes. 


Note  on  an  experiment  dealing  with  mutation  in  bacteria.  By 
L.  DoNCASTER,  Sc.D.,  King's  College. 

[Read  17  February  1919.] 

(Abstract.) 

It  was  noticed  that  the  recorded  ratio  of  occurrence  in  cases  of 
meningitis  of  the  four  agglutination-types  of  Meningococcus  corre- 
sponded very  closely  with  the  ratio  of  occurrence  of  the  four  iso- 
agglutinin  groups  of  blood  in  a  normal  human  population.  It 
seemed  possible,  therefore,  that  by  growing  Meningococcus  of  one 
type  in  media  containing  human  blood  of  ditferent  groups,  mutation 
to  other  types  might  be  induced.  Experiment  showed  that  con- 
siderable differences  in  type  of  agglutination  resulted,  but  it  was 
concluded  that  this  was  caused  by  the  sorting  out  of  races  of 
different  agglutinability  from  a  mass  culture,  rather  than  by  true 
mutation. 


CONTENTS. 

PAGE 

On  Certain  Trigonometrical  Series  lohich  have  a  Necessary  and  Sufficient 
Condition  for  Uniform  Convergence.  By  A.  E.  Jollifpe.  (Com- 
municated by  Mr  G.  H.  Hardy) 191 

Some  Geometrical  Interpretations  of  the  Concomitants  of  Ttvo  Qtiadrics. 

By  H.  ^Y.  TuRNBULL,  M.A.     (Communicated  by  Mr  G.  H.  Hardy)     196 

Some  properties  ofp{^n),  the  number  of  partitions  ofn.   By  S.  Ramanujan, 

B.A.,  Trinity  College 207 

Proof  of  certain  identities  in  combinatory  analysis  :  (1)  by  Professor 
L.  J.  Rogers;  (2)  by  S.- Ramanujan,  B.A.,  Trinity  College.  (Com- 
municated, with  a  prefatory  note,  by  Mr  G.  H.  Hardy)    .         .         .211 

On  Mr  Ramanujan's  congruence  properties  of  p  (n).    By  H.  B.  C.  Darling. 

(Communicated  by  Mr  G.  H.  Hardy) 217 

On  the  exponentiation  of  well-order^  series.   By  Miss  Dorothy  Wrinch. 

(Communicated  by  Mr  G.  H.  Hardy)        .         .         .        .         .        .219 

The  Gauss-Bonnet  Theorem  for  Midtiply -Connected  Regions  of  a  Siorfaee. 

By  Eric  H.  Neville,  M.A.,  Trinity  College 234 

On  an  empirical  formida  connected  with  GoldhacK's  Theorern.  By  N.  M. 
Shah,  Trinity  College,  and  B.  M.  Wilson,  Trinity  College.  (Com- 
municated by  Mr  G.  H.  Hardy) 238 

Note  on  Messrs  Shah  and  Wilson^s  pamper  entitled:  '■  On  cm  empirical 
formida  connected  xoith  GoldhacK s  Theory  \  By  G.  H.  Hardy,  M.  A., 
Trinity  College,  and  J.  E.  Littlewood,  M.A.,  Trinity  College         .     245 

The  distribution  of  Electric  Force  betioeen  tivo  Electrodes,  one  of  whixih  is 
covered  with  Radioactive  Matter.  By  "W.  J.  Harrison,  M.A.,  Fellow 
of  Clare  College 255 

The  conversion  of  soAV-diist  into  sugar.    By  J.  E.  Purvis,  M.A.       .        .    259 

BracTcen  as  a  sotorce  of  potash.     By  J.  E.  Purvis,  M.A 261 

The  action  of  electrolytes  on  the  electrical  conductivity  of  the  bacterial  cell 
and  their  effect  on  the  rate  of  migration  of  these  cells  in  an  electric 
field.    By  C.  Shearer,  Sc.D.,  F.R.S.,  Clare  College         .        .        .263 

The  bionomics  of  Aphis  grossiilariae  Kcdt.,  and  Aphis  viburni  Schr.  By 
Maud  D.  Haviland,  Bathm'st  Student  of  I^wnham  College.  (Com- 
municated by  H.  H.  Brindley) 266 

Note  on  an  expe7'iment  dealing  tvith  mutcUion  in  bacteria.  By  L.  Don- 
caster,  Sc.D.,  King's  College.     (Abstract)       .         .         .        .         ,     269 


* '  1  '^.^ 


PKOCEEDINGS 


OF  THE 


CAMBRIDGE  PHILOSOPHICAL 
SOCIETY 


VOL.   XIX.     PART  VI. 

[Easter  and  Michaelmas  Terms  1919.] 


(JDambttligt: 

AT  THE  UNIVERSITY  PRESS 

AND   SOLD  BY 
DEIGHTON,   BELL  &  CO.,    LIMITED, 
AND  BOWES  &  BOWES,  CAMBRIDGE. 

CAMBRIDGE  UNIVERSITY  PRESS, 
C.  F.  CLAY,  MANAGER,  FETTER  LANE,  LONDON,  E.G.  4 

1920 
Price  Three  Shillings  and  Siccpence  Net 
February  1920. 


NOTICES. 

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Transactions  may  be  had  on  application  to  Messrs  BowES  & 
Bowes  or  Messrs  Deighton,  Bell  &  Co.,  Limited,  Cambridge. 

3.  Other  volumes  of  the  Transactions  may  be  obtained  at 
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Society. 

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addressed  to  one  of  the  Secretaries,  ! 

Prof.  H.  F.  Baker,  St  John's  College.     [Mathematical.]  ! 

Mr  Alex.  Wood,  Emmanuel  College.     [Physical] 

Mr  H.  H.  Brindley,  St  John's  College.     [Biological] 

6.  Presents  for  the  Library  of  the  Society  should  be  ad- 
dressed to 

The  Philosophical  Library, 

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Diagrams  are  executed  as  far  as  possible  by  photographic  "process" 
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PROCEEDINGS 

OF   THE 


Colourimeter  Design.  By  H.  Hartridge,  M.D.,  Fellow  of 
King's  College,  Cambridge. 

[Received  7  October  1919;  read  10  November  1919.] 

In  a  previous  paper  (1)  I  have  described  certain  factors  which 
affect  the  efficiency  of  the  spectrophotometer.  The  colourimeter 
has  been  found  to  be  similarly  affected,  so  that  various  modifica- 
tions in  the  usual  designs  are  indicated. 

The  comparison  field  is  in  most  instruments  divided  at  a 
diameter,  so  that  one  half  receives  light  which  has  passed  through 
one  limb,  and  the  other  half  light  that  has  passed  through  the  other 
limb  of  the  instrument.  In  a  few  designs  the  bull's-eye  and  the 
central  strip  fields  have  been  employed.  All  these  fields  have 
the  disadvantage  that  local  stimulation  of  the  retina  may  occur  that 
sets  up  after  image  phenomena  greater  in  degree  in  one  part  than 
in  another,  thus  preventing  accurate  determinations.  And,  further, 
they  do  not  make  the  best  use  of  the  effects  of  simultaneous  contrast. 
A  better  type  of  field  is  the  one  which  I  have  previously  described 
in  connection  with  the  spectrophotometer,  namely,  one  which  is 
subdivided  into  a  number  of  strips,  of  which  alternate  numbers 
receive  light  from  the  two  limbs  of  the  instrument.  With  this  field 
the  eye  does  not  select  any  one  part  for  examination,  but  tends  rather 
to  judge  of  the  field  as  a  whole.  When  the  adjustment  of  intensity 
has  been  correctly  made  the  whole  field  should  become  uniform. 
The  effects  of  retinal  fatigue  therefore  tend  to  become  uniformly 
distributed.  The  contour  of  this  type  of  field  is  of  considerable 
length  compared  with  its  total  area ;  the  conditions  are  therefore 
beneficial  for  the  development  of  contrast.  The  absence  of  visible 
lines  of  junction  still  further  increases  this  effect. 

The  prisms  A  and  B  by  which  the  beams  of  light  through  the 
two  limbs  of  the  instrument  are  combined  at  the  compound  field 

VOL.   XIX.   PART  VI.  19 


272 


Br  HarLridge,  Colourimeier  Design 


Eyepiece  Cup  - 


Ramsden  eyepiece  • 


Comparison  field 


Comparison  field 


Plunger  ■ 


Staa;e  - 


Tail  piece  to  carry  lamp 


f*ush-on  terminals 
Horse-shoe  foot 


Opal  glass  disc 
Watt  lamp 


Dr  Hartridge,  Colourimeter  Design  273 

described  above,  are  similar  in  shape  to  those  used  in  the  spectro- 
photometer. They  are  shown  in  the  diagram  of  the  apparatus.  It 
will  be  observed  that  the  interface  of  the  prisms  is  silvered,  the 
metallic  film  being  removed  by  means  of  a  simple  ruling  machine,  so 
that  narrow  strips  of  the  silver  alternate  with  strips  from  which  the 
whole  of  the  silver  has  been  removed.  Examination  of  the  diagram 
will  show  that  by  this  arrangement  the  field  seen  on  looking  down  the 
eyepiece  is  formed  of  alternating  narrow  beams  which  have  either 
been  transmitted  from  one  limb  of  the  instrument  through  the 
spaces  between  the  silver  strips,  or  reflected  from  the  other  limb 
by  the  silver  strips  themselves.  The  lengths  of  the  prisms  A  and 
JB  should  be  such  that  the  two  entering  beams  have  passed  through 
equal  lengths  of  glass. 

The  troughs  are  adjustable  on  both  limbs  of  the  instrument, 
in  colourimeters  of  usual  design.  This  arrangement  has  the  dis- 
advantage that  if  there  should  be  any  backlash  in  the  micrometer 
mechanism  which  is  used  for  adjusting  the  position  of  the  movable 
troughs,  or  error  in  the  setting  of  the  scale,  these  will  affect  both 
the  thickness  of  the  pigment  solution  to  be  estimated,  and  also 
that  of  the  standard.  Such  errors  can  be  eliminated  so  far  as  the 
standard  is  concerned  by  the  use  of  a  special  cell,  the  distances 
between  the  sides  of  which  are  determined  by  accurately  ground 
distance  pieces,  which  may  be  made  of  either  glass  or  metal. 
Rustless  steel  would  appear  to  be  a  suitable  metal  because  it  resists 
the  corrosive  action  of  ordinary  solvents. 

I  have  shown  that  in  the  case  of  the  spectrophotometer  there 
are  important  reasons  for  the  use  of  troughs  with  double  compart- 
ments on  both  limbs  of  the  instrument.  In  both  ti'oughs  the  com- 
partment near  the  light  source  should  contain  the  solvent  only, 
the  other  being  filled  with  the  solution  of  the  pigment.  Double 
troughs  should  be  used  with  the  colourimeter  for  similar  reasons, 
namely,  (a)  in  order  that  absorption  by  the  solvent  may  be  com- 
pensated, since  the  thickness  is  the  same  on  both  sides  of  the 
instrument ;  (b)  that  pigments  accompanying  the  one  under  esti- 
mation may  be  compensated  for;  (c)  that  specific  surface  reflection 
at  the  sides  of  the  troughs  which  contain  pigment  may  be  similar 
on  both  limbs  of  the  instrument.  With  regard  to  the  type  of 
trough  that  should  be  employed  I  have  previously  considered  the 
advantages  of  the  double  wedge  trough  in  conjunction  with  the 
spectrophotometer.  In  the  case  of  the  colourimeter  the  plunger 
type  usually  employed  has  the  advantage  of  not  requiring  calibra- 
tion with  a  micrometer  microscope  as  wedge  troughs  do.  The 
method  of  employing  double  compartment  plunger  troughs  and 
standard  troughs  is  shown  in  the  diagram.  In  some  colourimeters 
the  troughs  are  bell  mouthed,  and  are  manufactured  from  black 
glass.    These  points  are  to  be  recommended.    It  should  be  noted, 

19—2 


274  Dr  Hartridge,  Colourimeter  Design 

however,  that  reflection  can  still  take  place  at  the  sides  of  the 
troughs,  so  that  it  is  necessary  carefully  to  restrict  the  light  illu- 
minating the  troughs  to  narrow  vertical  pencils  of  just  sufficient 
diameter  fully  to  illuminate  the  comparison  fields.  Since  scattered 
or  reflected  light  may  increase  the  apparent  brightness  of  one  of 
the  fields  it  is  essential  that  this  be  reduced  to  a  minimum.  Special 
care  should  therefore  be  taken  in  designing  the  instrument  to  pre- 
vent the  entrance  of  stray  light,  and  to  employ  an  illuminating 
system  that  will  limit  the  entering  beams  to  the  narrow  pencils 
above  referred  to. 

The  illumination  in  the  majority  of  colourimeters  is  obtained 
from  the  sky  by  means  of  a  plane  mirror.  In  some  instruments  this 
may  be  replaced  at  will  by  a  finely  matted  white  surface.  The 
illumination  therefore  in  either  case  consists  of  a  large  number  of 
divergent  pencils,  which  enter  the  lower  ends  of  the  troughs  in  all 
possible  directions.  Scattered  light  is  therefore  at  a  maximum.  In 
the  case  of  the  microscope  a  similar  practice  used  to  be  in  vogue,  but 
it  has  given  way  to  the  use  of  illuminating  lens  systems  in  which 
the  corrections  and  alignment  are  well  nigh  as  perfect  as  those 
used  in  the  objective  and  eyepiece.  Now,  in  the  case  of  the  spec- 
trophotometer I  have  shown  that  the  beams  illuminating  the  two 
limbs  of  the  instrument  should  proceed  from  identical  parts  of  the 
light  source.  This  condition  should  be  realised  in  the  case  of  the 
colourimeter  also.  The  arrangement  of  the  illuminating  apparatus 
is  shown  in  the  diagram. 

The  light  source  is  similar  to  that  which  I  have  applied  to  the 
microscope  (2),  consisting  of  a  slab  of  white  opal  glass  finely 
ground  on  both  sides.  This  is  lit  from  behind  by  means  of  a  small 
half  watt  electric  lamp,  which  obtains  its  current  from  a  small 
accumulator  or  dry  cell,  or  from  the  town  supply  through  a  suit- 
able resistance.  The  lamp  is  enclosed  in  a  brass  box,  which  is 
silver  plated  inside,  and  is  finished  dead-black  outside  so  as  to 
radiate  heat.  The  life  of  the  lamp  is  increased  by  connecting  it 
with  a  press  switch  so  that  it  is  in  circuit  during  observation  only. 
The  lamp  box  is  attached  to  the  tail-piece  of  the  instrument  so 
that  it  forms  an  integral  part  of  the  apparatus.  The  whole  may 
thus  be  tilted  or  moved  from  place  to  place  without  requiring  re- 
adjustment. Immediately  above  the  opal  glass  is  a  metal  dia- 
phragm, the  aperture  in  which  limits  the  surface  exposed  to  a 
disc  4  mm.  in  diameter.  Attached  beneath  the  stage  of  the  in- 
strument and  60  mm.  above  the  diaphragm  of  the  light  source  is 
a  plano-convex  achromatic  lens  of  26  mm.  diameter  and  60  mm. 
focal  length.  The  divergent  rays  from  each  point  of  the  source 
are  rendered  parallel  by  this  lens,  and  at  once  pass  through  two 
achromatic  plano-convex  lenses  of  18  cms.  focal  length  and  14  mm. 
diameter.    These  lenses  have  a  clear  aperture  of  12  mm.  and  form 


Dr  Harfridge,  Colourimeter  Design  275 

a  focussed  image  of  the  diaphragm  of  the  light  source,  which  is 
magnified  in  the  ratio  of  the  focal  lengths  of  the  lenses;  since  the 
ratio  is  3  to  1  this  image  has  a  diameter  of  12  mm. 

The  beams  that  emerge  through  the  lenses  Tl  and  T'2  do  not 
therefore  anywhere  exceed  12  mm.  and  the  light  does  not  spread, 
for  this  reason,  to  the  sides  of  the  troughs  during  its  passage 
and  therefore  stray  light  is  reduced  to  a  minimum.  The  beam 
from  the  lens  T2  passes  vertically  upwards  through  a  hole  in 
the  stage  to  the  standard  trough  which  rests  upon  it.  Having 
passed  through  both  the  layer  of  solvent  and  also  that  of  the 
solution  of  pigment,  the  beam  enters  prism  B',  and  is  totally  inter- 
nally reflected  at  its  inclined  surface  on  to  the  silvered  strips  of  the 
comparison  field.  The  beam  that  has  passed  through  Tl  is  deflected 
by  internal  reflection  at  the  right  angled  prism  C  which  is  cemented 
to  it,  and  falls  on  the  silvered  sui'face  between  the  two  halves  of  the 
prism  D,  so  that  the  beam  is  directed  vertically  through  a  second 
hole  in  the  stage  on  to  the  lower  fixed  cup  of  the  adjustable  trough, 
which  is  filled  with  solvent.  It  then  passes  through  the  movable 
cup  which  contains  the  pigment,  and  enters  the  prism  A  to  fall  on 
the  silvered  strips  of  the  comparison  field.  The  passage  of  this 
beam  through  the  intervals  between  the  strips,  and  the  reflection 
of  the  beam  from  the  other  limb  of  the  instrument  at  the  strips 
themselves,  has  already  been  described.  It  will  be  noted  that  the 
reflection  of  the  one  beam  by  internal  reflection  within  the  prism 
C,  and  by  ordinary  reflection  within  the  prism  D,  causes  this  beam 
to  compensate  for  the  internal  reflection  and  reflection  at  a  silvered 
surface  which  occurs  within  prism  B  in  the  case  of  the  other  beam. 
As  it  has  been  found  that  silvered  surfaces  vary  in  the  intensity 
of  rays  of  different  wave-length  which  they  reflect,  it  is  advisable 
that  both  mirror  D  and  prism  B  be  silvered  with  the  same  solution 
at  the  same  time. 

The  lengths  of  the  paths  of  the  beams  through  the  instrument 
are  found  to  be  in  the  case  of  the  left-hand  beam  an  actual  dis- 
tance of  19'5  cms.,  that  is  an  effective  distance  of  18  cms.  since 
2"2  cms.  of  glass  is  passed  through ;  in  the  case  of  the  right-hand 
beam  the  total  and  the  equivalent  lengths  are  the  same  as  those 
on  the  left. 

The  comparison  field  therefore  is  illuminated  by  two  super- 
posed images  of  the  diaphragm  of  the  light  source,  one  of  which 
has  passed  through  the  standard  trough  and  the  other  through  the 
adjustable  trough.  When  the  instrument  is  in  correct  adjustment 
these  two  images  exactly  coincide,  so  that  if  there  should  be  any 
slight  inequality  between  the  intensity  of  illumination  of  different 
parts  of  the  light  source  both  images  will  be  similarly  affected, 
and  therefore  the  match  between  their  different  parts  will  remain 
unchanged.    Such  a  condition  is  not  secured  in  the  usual  forms  of 


276  Br  Hartridge,  Colourimeter  Design 

colourimeter,  since  it  is  due  to  the  particular  method  of  illumina- 
tion described  above. 

The  eyepiece  used  in  the  du  Bosq  type  of  colourimeter  consists 
of  a  Eamsden  lens  system,  at  the  upper  focal  plane  of  which  has 
been  placed  a  diaphragm  pierced  with  a  small  aperture.  This  has 
the  effect  of  limiting  the  rays  reaching  the  eye  to  those  which 
have  passed  as  approximately  parallel  bundles  up  the  limbs  of  the 
instrument.  To  be  effective  the  aperture  has  to  be  small,  and  this 
has  the  disadvantage  of  making  the  intensity  of  illumination  of 
the  fields  somewhat  low.  When  this  type  of  eyepiece  is  in  use  it 
is  found  that  the  eye  has  to  be  inconveniently  close  to  the  aperture 
in  order  that  the  whole  field  shall  be  seen  at  one  and  the  same 
time.  This  is  due  to  the  fact  that  the  diaphragm  is  a  considerable 
distance  below  the  effective  pupil  of  the  eye,  even  when  the  eye 
has  been  placed  as  close  as  possible,  and  as  a  result  some  of  the 
rays  which  spread  out  from  the  diaphragm  may  not  enter  the  pupil. 
The  difficulty  is  in  fact  similar  to  that  met  with  in  high  power 
microscopic  eyepieces  of  the  Huygenian  type.  To  avoid  this  diffi- 
culty a  more  elaborate  type  of  eyepiece  has  been  devised,  in  which 
an  erecting  lens  system  has  been  placed  above  the  Ramsden  ocular 
and  its  diaphragm  (8).  This  causes  a  sharp  image  to  be  seen  on 
looking  down  the  eyepiece,  and  at  the  same  time  the  image  of  the 
small  aperture  is  formed  at  a  considerable  distance  above  the  top 
lens,  so  that  the  eye  does  not  have  to  be  placed  inconveniently 
close  to  the  eyepiece  in  order  to  obtain  a  full  view  of  the  field. 
These  improvements  are  obtained,  however,  at  a  certain  sacrifice 
of  definition,  which  is  unimportant  in  the  usual  types  of  colouri- 
meter in  which  the  fields  are  of  simple  design,  but  is  of  relatively 
greater  importance  if  the  more  detailed  type  of  field  be  used  which 
has  been  described  above.  It  will  have  been  observed  that  in  the 
colourimeter  which  I  have  described  above  the  illuminating  beams 
are  formed  by  the  special  method  of  illumination  employed.  Under 
which  circumstances  it  is  found  that  the  Ramsden  disc  of  the  ocular 
contains  the  overlapping  focussed  images  of  the  restricting  aper- 
tures of  the  lenses  Tl  and  T2,  which  when  the  instrument  is  in 
correct  adjustment  exactly  overlay  one  another.  It  is  therefore 
unnecessary  that  the  eyepiece  should  contain  any  diaphragm  to 
restrict  the  beams,  and  therefore  the  difficulties  introduced  by 
such  a  diaphragm  are  not  met  with.  The  eyepiece  itself  should 
be  achromatic  and  should  slide  in  a  tight-fitting  jacket  so  that  the 
observer  may  set  it  at  the  best  focus.  It  should  magnify  about  3 
diameters. 

The  angle  at  which  the  comparison  field  lies  will  be  seen  to  be 
45  degrees.  But  since  it  is  enclosed  between  two  pieces  of  glass, 
the  apparent  angle  to  the  eye  is  reduced  in  the  ratio  of  the  refi'ac- 
tive  indices  of  glass  and  air.    The  apparent  angle  would  therefore 


Dr  Hartridge,  Colourimeter  Design  277 

be  about  29  degrees.  Now,  the  dimensions  of  the  field  seen  by  the 
eye  are  8  mm.  by  6  mm.,  the  latter  being  in  the  direction  of  the 
slope.  The  apparent  diiference  of  focus  is  therefore  less  than  4  mm., 
which  would  be  equivalent  to  12  mm.  at  a  distance  of  25  cms. 
Such  a  small  change  of  focus  would  be  at  once  met  by  a  trifling 
change  in  the  degree  of  accommodation  of  the  eye,  which  would 
be  effected  subconsciously  and  involuntarily.  No  difficulty  is  to  be 
met  with  therefore  from  this  cause. 

The  Mechanical  System. 

The  metal  work  of  the  colourimeter  follows  closely  that  of  the 
microscope.  The  horse-shoe  foot,  stage  and  coarse  adjustment  all 
resemble  those  used  in  that  instrument.  The  adjustment  has  a 
range  of  40  mm.  only,  because,  as  will  be  shown  later,  the  use  of 
standard  solutions  of  20  mm.  thickness  makes  a  bigger  movement 
than  this  unnecessary.  An  accuracy  of  one-quarter  per  cent,  should 
be  sufficient,  and  this  is  readily  provided  by  a  scale  gi^aduated  in 
half  mm.  and  reading  by  a  vernier  to  one-twentieths.  The  adjust- 
ment should  have  long,  well-made  V  slides  so  as  to  eliminate  lost 
motion.  The  scale  should  be  attached  to  the  moving  member,  the 
vernier  being  attached  to  the  fixed.  A  simple  lens  and  45  degree 
mirror  should  make  a  magnified  image  of  this  visible  to  the  ob- 
server. To  the  moving  member  is  first  screwed  and  afterwards 
sweated  with  soft  solder  a  strong  brass  ring.  To  this  is  attached 
by  means  of  a  three-prong  bayonet  catch  the  ring  fixed  to  the 
upper  lip  of  the  movable  trough.  The  trough  is  cemented  into  a 
groove  turned  in  this  ring  by  means  of  plaster  of  Paris  or  Caemen- 
tium.  Where  plaster  has  been  used  the  joint  should  be  covered 
by  a  thin  coat  of  Robiallac.  The  prisms  and  eyepiece  are  attached 
to  a  strong  projection  at  the  top  of  the  pillar  which  forms  the 
handle  of  the  instrument. 

The  removal  of  the  troughs  for  filling  and  cleaning  and  their 
replacement  is  a  simple  process  which  should  not  take  more  than 
a  few  seconds.  To  remove  the  adjustable  troughs,  first  swing  the 
substage  to  one  side ;  this  allows  the  lower  trough  to  drop  verti- 
cally through  the  hole  in  the  stage  until  it  can  be  removed.  The 
upper  trough  is  now  gripped  between  the  finger  and  thumb,  and 
the  trough  rotated  so  as  to  free  the  bayonet  catches ;  this  trough 
is  then  lowered  through  the  hole  in  the  stage  and  removed.  The 
plunger  and  the  troughs  can  now  be  cleaned,  refilled  and  returned. 
The  standard  double  trough  simply  rests  on  its  side  of  the  stage, 
so  that  its  removal  takes  but  a  moment. 


278  Dr  Hartridge,  Colourimeter  Design 

The  Colourimeter  in  Practice. 

Experiment  has  shown  that  if  two  solutions  of  the  same  colour 
contain  different  pigments  in  solution,  then  the  thicknesses  re- 
quired for  a  match  vary  not  only  with  the  observer  and  with  the 
quality  of  the  light,  but  also  with  the  same  observer  from  time  to 
time.  It  is  for  this  reason  that  the  technique  has  been  introduced 
of  using  the  same  pigment  for  the  standard  as  that  required  to  be 
estimated.  Thus  creatinin  is  no  longer  estimated  by  comparing  the 
colour  which  develops  when  picric  acid  and  soda  are  added  with  the 
colour  of  a  solution  of  potassium  dichromate ;  but  a  standard  solution 
of  creatinin  is  used,  picric  acid  being  added  to  it  at  the  same  time  as 
it  is  added  to  the  solution  to  be  standardised.  If,  then,  the  thick- 
ness of  the  standard  is  20  mm.  and  that  of  the  unknown  17  mm., 
it  is  assumed  that  the  strengths  of  the  solutions  are  in  the  inverse 
ratio  of  those  numbers.  Such  is  not  the  case  however,  because 
the  sodium  picrate  itself  absorbs  rays  from  the  same  part  of  the 
spectrum  as  does  the  sodium  picramate,  and  therefore,  although 
the  light  may  encounter  the  same  number  of  coloured  radicals  in 
both  limbs  of  the  instrument,  yet  the  sodium  picrate  absorption  is 
greater  on  one  side  than  the  other,  because  the  fluids  are  not  of 
the  same  thickness.  It  is  principally  for  this  reason  that  I  have 
adopted  an  instrument  in  which  double  troughs  are  used,  on  both 
sides  of  the  instrument ;  the  lower  pair  on  both  sides  being  filled 
with  sodium  picrate  solution  in  the  case  taken  above  as  example,  the 
upper  pairs  containing  the  picric  acid  plus  creatinin.  In  this  way 
the  number  of  picrate  radicals  is  kept  approximately  constant,  since 
the  total  thickness  of  sodium  picrate  solution  is  the  same  on  both 
sides  of  the  instrument.  The  balance  is  not  perfect  however,  because 
a  certain  amount  of  picric  acid  is  used  up  in  forming  the  sodium 
picramate,  and  this  amount  cannot  be  ascertained  without  assum- 
ing that  the  estimation  to  be  done  has  already  been  accurately 
performed.  The  problem  is,  in  fact,  represented  by  a  simultaneous 
equation  involving  two  unknowns.  I  find  that  the  matter  can  be 
solved  in  the  following  manner.  Having  diluted  both  the  standard 
and  the  unknown  solutions  with  equal  amounts  of  standard  picric 
acid  and  soda  solutions,  and  having  allowed  the  colour  to  develop  in 
the  ordinary  manner,  an  estimate  is  made  of  the  relative  strengths  of 
the  solutions  in  the  colourimeter.  Having  found  that,  say,  a  20  mm. 
thickness  of  the  standard  has  the  same  tint  as  13"4  mm.  of  the 
unknown  solution,  a  fresh  sample  of  the  unknown  is  taken  and 
13'4  c.c.  of  it  diluted  with  water  to  bring  the  total  to  20  c.c.  The 
solution  of  the  unknown  has  thus  been  brought  to  approximately 
the  same  concentration  as  the  standard.  (Where  the  approximate 
strength  is  known  a  preliminary  dilution  before  making  the  initial 
estimation  is  beneficial.)    The  correctly  diluted  solution  of  the  un- 


Dr  Hartridge,  Colourimeter  Design  279 

known  is  now  treated,  ab  initio,  with  fresh  picric  acid  solution  and 
soda,  and  is  then  estimated  against  the  standard  in  the  colourimeter. 
It  is  now  found  that  a  20  mm.  thickness  of  the  standard  has  the 
same  tint  as  one  of,  say,  19"85  of  the  unknown  after  dilution.  The 
strength  of  the  unknown  is  thus  ascertained,  with  considerable 
accuracy,  because  the  conditions  of  equilibrium  under  which  the 
sodium  picramate  develops  and  exists,  and  the  quantities  of  picric 
acid  used  up  in  the  determination  are  approximately  constant. 

It  should  be  pointed  out  that  the  above  technique  presents  no 
difficulties,  and  takes  little  longer  than  the  ordinary  method.  The 
p)rinciple  may  with  advantage  be  applied  to  all  estimations  made 
Avith  the  colourimeter. 

The  Accuracy  of  the  Colourimeter. 

Since  colour  is  due  to  absorption  the  colourimeter  depends  for 
its  utility  on  the  fact  that  a  change  in  the  number  of  coloured 
radicals  encountered  by  light  causes  a  change  in  the  retinal  stimu- 
lus when  that  light  falls  on  the  eye.  We  may,  therefore,  arbitrarily 
state  that  the  accuracy  of  the  determinations  depends,  firstly,  on 
the  rate  of  change  in  the  quality  of  the  light  which  is  passed 
through  the  pigment,  and,  secondly,  on  the  acuteness  of  the  per- 
ception of  the  eye  for  the  change  in  quality  of  the  light.  The 
greater  the  rate  of  change  and  the  greater  the  acuteness  of  percep- 
tion of  that  change,  the  greater  will  be  the  accuracy.  Many  bodies 
which  absorb  light  do  so  selectively,  that  is,  they  have  a  gref^ter 
effect  in  one  part  of  the  spectrum  than  in  another ;  they  therefore 
show  colour,  that  is,  they  are  pigments.  Under  ordinary  circum- 
stances the  greater  the  absorption  the  stronger  the  colour  and  the 
less  the  intensity  of  the  transmitted  light.  As  the  concentration 
of  a  pigment  is  altered,  and  therefore  the  degree  of  absorption,  the 
strength  of  colour  and  the  brightness  of  the  transmitted  light  both 
vary.  The  colourinietric  determination,  therefore,  depends  on  the 
simultaneous  occurrence  of  both  these  changes.  The  important 
questions  that  arise  are :  (1)  on  what  do  the  magnitudes  of  these 
changes  depend  ?  (2)  which  is  the  more  important  ?  and  (3)  how 
can  the  changes  be  increased  for  a  given  alteration  in  concentra- 
tion ?  A  study  of  absorption  band  formation  gives  a  definite  answer 
to  each  of  these  questions  as  follows:  (1)  The  changes  for  a  given 
alteration  of  concentration  are  greater  the  flatter  and  broader  the 
absorption  band.  If,  therefore,  there  were  two  pigments  of  the  same 
concentration  and  the  same  colour,  i>ne  of  which  had  a  sharp  well- 
defined  band,  while  that  of  the  other  was  broad  and  flat,  the  latter 
pigment  would  be  found  to  give  the  more  accurate  readings  in  the 
colourimeter.  (2)  Of  the  two  changes,  that  of  colour  is  usually  the 
more  important,  particularly  with  pigments  showing  single  absorp- 
tion bands.    In  pigments  with  multiple  bands  the  intensity  change 


280  Dr  Hartridge,  Colourimeter  Design 

may  be  the  more  important :  for  example,  a  pigment  absorbing  to 
an  equal  extent  in  two  complementary  parts  of  the  spectrum  will 
cause  the  light  to  suffer  no  change  in  colour  at  all,  while  the  in- 
tensity is  altered.  (3)  The  changes  in  the  case  of  any  one  pigment 
can  be  increased  by  increasing  the  intensity  of  that  part  of  the 
spectrum  which  is  suffering  change  or  by  decreasing  that  of  parts 
which  do  not  show  alteration.  Of  the  two  methods  the  latter  is 
the  easier  to  carry  out  and  the  more  efficient.  If  colour  filters  are 
used  they  must  be  carefully  adjusted  according  to  the  position  in 
the  spectrum  of  the  absorption  band  of  the  pigment  to  be  estimated. 
If  a  spectral  illuminator  is  used  the  apparatus  virtually  becomes 
a  spectrophotometer,  and  this  elaboration  is  hardly  necessary  for 
ordinary  work.  The  possibility  should  not  be  overlooked  of  the 
existence  of  alternative  colour  reactions  to  those  at  present  in  use 
in  which  pigments  having  less  steep  absorption  bands  are  used  and 
which  therefore  permit  greater  accuracy  in  their  colourimetric 
estimation. 

The  factors  which  influence  the  acuteness  of  perception  of  the 
eye  remain  for  consideration.  Firstly,  it  is  clear  since  the  accuracy 
of  the  determination  depends  on  the  correctness  of  the  match  ob- 
tained, that  the  eye  should  not  be  suffering  from  fatigue.  The 
reading  of  small  print  and  the  exposure  of  the  eyes  to  excessive 
light  should,  therefore,  be  avoided  for  a  reasonable  time  before  the 
determinations.  The  absence  of  refractional  errors,  eye  strain,  want 
of  eye-muscle  balance  and  the  possession  of  good  general  health  are 
all  factors  of  importance.  In  my  own  case  the  period  after  tea  is  the 
best,  provided  that  the  morning's  work  has  not  been  arduous.  The 
presence  of  after  images  is  most  harmful  for  accurate  estimations ; 
the  best  method  of  eliminating  them  is,  I  find,  to  look  for  a  fcAV 
moments  at  a  uniformly  lit  grey  surface.  All  the  above  points  may 
seem  obvious ;  it  is  however  my  experience  to  find  that  they  are 
sometimes  overlooked.  The  apparatus  itself  is  best  placed  in  a  dark 
room,  or  at  all  events  where  the  full  light  of  a  window  cannot  fall 
on  the  eye  of  the  observer.  In  the  latter  case  the  eyepiece  cup 
may  be  made  deep  with  advantage,  so  as  to  protect  the  periphery 
of  the  retina  from  stimulation  and  thus  bring  about  an  increase  in 
the  diameter  of  the  pupil. 

With  regard  to  the  use  of  colour  filters,  experiment  shows  that 
the  theoretical  conclusions  arrived  at  above  are  amply  justified, 
namely,  that  the  accuracy  of  the  determinations  is  increased  if 
either  the  rays  absorbed  by  the  pigment  are  increased  in  intensity, 
or  those  not  absorbed  are  decreased  or  removed  altogether.  The 
removal  by  means  of  colour  filters  is  however  usually  attended  by  so 
great  a  diminution  in  the  intensity  of  the  light  that  a  powerful 
source  such  as  an  arc  lamp  becomes  necessary.  It  is  a  fortunate 
circumstance,  therefore,  that  the  retina  should  be  even  more  sensi- 


Dr  Hartridge,  Colourimeter  Design  281 

tive  to  change  in  shade  than  it  is  to  change  in  intensity.  I  have 
found,  further,  that  the  point  of  greatest  sensitiveness  is  obtained 
when  the  fields  are  nearly  neutral  in  colour.  Such  a  condition  is 
obtained  by  the  use  of  a  suitable  colour  filter  which  absorbs  in  that 
part  of  the  spectrum  which  is  occupied  by  the  complementary 
colour  to  that  absorbed  by  the  pigment.  Suppose,  for  example,  a 
yellow  pigment  is  to  be  estimated,  then  a  blue  solution  of  a  dye  is 
placed  in  the  path  of  the  light  from  the  source  of  such  a  thickness 
and  concentration  that  the  comparison  field  seen  in  the  instrument 
is  of  a  neutral  grey  colour.  Permanent  colour  films  between  glass 
should  be  used  if  much  work  is  likely  to  be  done  with  any  given 
pigment.  Such  a  technique  is  very  simple,  and  I  find  that  in  my 
hands  it  increases  the  accuracy  of  the  determinations  by  about 
three  times  (when  estimating  sodium  picrate),  the  method  of  mean 
squares  being  used  to  calculate  the  average  error  of  the  experi- 
mental determinations  both  with  and  without  the  complementary 
filter.  The  probable  error  of  the  determinations  was  found  to  be 
0"8  per  cent.,  using  home-made  apparatus  and  the  complementary 
screen.  It  should  be  possible  to  halve  this  amount  if  the  precau- 
tions outlined  above  be  taken  and  well-designed  apparatus  be  used. 

Summary. 

(1)  The  comparison  field  seen  on  looking  down  the  instrument 
should  cause  the  greatest  contrast  and  at  the  same  time  should  not 
produce  after  images. 

(2)  On  both  limbs  of  the  instrument  double  troughs  should  be 
used,  so  that  the  thickness  of  pigment  to  be  measured  may  be 
varied  at  will,  while  the  absorption  caused  by  other  pigments 
remains  constant. 

(3)  An  artificial  light  source  should  be  used,  and  the  lighting 
system  be  so  designed  that  narrow  beams  are  produced  of  just 
sufficient  width  as  to  completely  illuminate  the  comparison  field. 
The  amount  of  reflected  and  scattered  light  may  thus  be  reduced 
to  a  minimum. 

(4)  If  experiment  shows  that  the  change  in  colour  produced 
by  a  given  change  in  thickness  or  concentration  of  the  pigment 
can  be  increased  by  modifying  the  relative  intensity  of  different 
parts  of  the  spectrum  of  the  light  source,  then  suitable  colour  filters 
should  be  prepared  for  use  during  the  determinations.  It  was 
found  in  a  test  case  that  this  modification  alone  increased  the 
accuracy  by  three  times. 

(5)  The  general  design  of  the  instrument  should  conform  to 
microscopic  practice,  fixed  troughs  being  supported  by  the  stage 
and  the  movable  trough  actuated  by  the  rack  and  pinion  course 


282  Dr  Hartridge,  Colourimeter  Design 

adjustment  screw.  The  illuminating  system  should  be  fitted 
beneath  the  stage  so  that  the  instrument  may  be  tilted  or  moved 
from  place  to  place  without  disturbing  the  alignment. 

For  certain  purposes  it  may  be  found  beneficial  to  employ 
smaller  quantities  of  liquid  than  those  required  in  the  ordinary 
colourimeter.  I  find  that  a  modification  in  the  design  of  the 
troughs  should  make  1  to  2  c.c.  of  liquid  sufficient ;  and  further,  by 
modifying  the  optical  system  as  well,  as  little  as  "001  c.c.  could  be 
worked  with.  It  should  be  pointed  out  however  that  such  quantities 
could  only  be  employed  with  solutions  of  considerably  greater  con- 
centration than  those  usually  estimated ;  e.g.  about  ten  times  the 
usual  concentration  for  1  c.c,  and  one  hundred  times  for  '001  c.c. 

References. 

(1)  Hartridge,  Journ.  Physiol,  l,  p.  101  (1915). 

(2)  Hartridge,  Joum.  Qitekett  Micro.  Soc.  Nov.  1919. 

(3)  Kober,  Journ.  Biol.  Ghem.  xxix,  p.  155  (1917). 


Mr  Snell,  The  Natural  History  of  the  Island  of  Rodrigues    283 


The  Natural  History  of  the  Island  of  Rodrigues.  By  H.  J.  Snell 
(Eastern  Telegraph  Company)  and  W.  H.  T,  Tams^.  (Communi- 
cated by  Professor  Stanley  Gardiner.) 

[Read  10  November  1919.] 

Rodrigues  lies  some  350  miles  east  of  Mauritius,  and  is  a  rugged 
mass  of  volcanic  rock  closely  resembling  Mauritius  and  Reunion. 
It  is  surrounded  by  a  coral  reef,  the  edge  of  which  at  the  eastern 
end  is  within  100  yards  of  the  beach,  whilst  on  the  north  and  south 
it  extends  outwards  to  a  distance  of  three  to  four  miles,  and  on  the 
west  to  two  miles.  There  is  an  irregular  channel  inside  the  reef 
close  to  the  shore,  extending  round  most  of  the  island,  sufficiently 
deep  for  boats  at  any  state  of  the  tide,  and  at  the  south-east  end 
a  small  lagoon  of  three  to  ten  fathoms,  with  a  passage  through 
the  reef.  The  usual  anchorage  is  Mathurin  Bay,  in  the  reef  to  the 
north.  The  reef  is  studied  with  islets,  those  nearer  the  shore  being 
mostly  of  volcanic  nature,  and  situated  on  the  north  and  west, 
whilst  the  rest  are  of  limestone,  modern  accumulations  of  debris, 
and  situated  on  the  south. 

The  island  itself  is  eleven  miles  long  by  five  miles  broad,  and 
has  an  area  of  just  over  forty  square  miles.  There  is  a  central 
lofty  ridge  extending  from  east  to  west,  with  a  break  about  one- 
third  of  its  length  from  the  west.  The  western  bastion  of  the  range 
is  Mount  Quatre- Vents,  1120  feet  high,  while  at  the  eastern  end 
is  Grande  Montaigne,  1140  feet.  The  highest  point  is  Mount 
Limon  (1300  feet),  which  lies  with  two  other  peaks  a  little  out  of 
the  general  line  of  mountains.  The  sides  of  these  peaks  are  cut 
into  numerous  ravines,  these  being  deeper  and  more  frequent  on 
the  south  side  than  on  the  north.  At  their  upper  ends  these  ravines 
are  often  bordered  by  perpendicular  columnar  basaltic  cliffs, 
sometimes  exceeding  200  feet  in  height,  extensively  cut  into  many 
coulees  by  small  streams  which  often  descend  in  a  series  of  cascades. 

The  volcanic  ridge  descends  on  the  south-west  gradually,  and 
passes  into  a  broad  coralHne  limestone  plain,  with  occasional  hills 
up  to  500  feet  high,  indicating  a  comparatively  recent  elevation 
of  at  least  a  like  amount.  This  tract  of  limestone  is  honeycombed 
with  caves,  in  which  stalactites  and  stalagmites  are  abundant. 
There  are  many  holes  and  fissures,  and  often  deep  hollows  occur, 
at  the  bottom  of  which  lie  large  fragments  of  limestone  in  irregular 
heaps;  these  are  apparently  old  caves,  the  roofs  of  which  have 
fallen  in.  The  floors  of  these  hollows  are  covered  with  soil,  often 

1  The  second  author  is  solely  responsible  for  the  names  of  the  insects  herein 
recorded. 


284    Mr  Snell,  The  Natural  History  of  the  Island  of  Rodrigues 

witli  lumps  of  volcanic  rock  on  the  surface.  The  limestone  is  not 
found  along  the  northern  or  southern  shores,  except  at  their  eastern 
extremity,  where  patches  occur  at  the  mouths  of  the  valleys, 
occasionally  at  some  distance  from  the  shore.  Some  of  the  patches 
of  limestone  found  in  the  volcanic  region  indicate  an  elevation  of 
perhaps  500  feet,  and  the  raised  beaches  on  the  south  shore,  some 
20  feet  in  height,  may  point  to  a  further  subsequent  change  of 
level.  The  position  of  old  volcanic  craters  has  not  been  accurately 
determined,  but  the  main  ones  appear  to  have  been  situated 
about  the  Grande  Montaigne  and  Mount  Malartic, 

The  island  is  comparatively  dry,  and  during  the  warm  season 
many  of  the  streams  are  dried  up,  though  they  assume  in  the 
rainy  season  torrential  proportions.  The  climate  is  like  that  of 
Mauritius.  The  rainfall  is  very  irregular;  during  the  north-west 
monsoon  from  November  to  April  the  weather  is  wet  and  warm, 
and  early  in  this  season  there  are  frequently  severe  hurricanes. 
From  May  to  October  the  south-east  monsoon  prevails,  and  the 
weather  is  then  cool  and  dry.  Fogs  are  rare,  and  climatic  conditions 
render  the  island  healthy  to  live  in. 

Rodrigues  was  discovered  in  1510,  by  a  Portuguese  commander, 
whose  name  it  bears.  In  1691  the  Dutch  landed  several  fugitive 
French  Huguenots  there,  among  whom  was  M.  Fran9ois  Leguat, 
who  wrote  an  account  of  the  island  in  1708.  The  island  was  later 
cultivated  by  the  French  East  Indian  Company,  and  maize  and 
corn  were  grown ;  these,  with  dried  fish,  turtles  and  land  tortoises, 
were  exported  to  Mauritius.  It  was  occupied  by  the  British  in 
1809,  and  made  the  base  of  operations  against  Mauritius.  It  is 
still  cultivated  as  a  garden  for  Mauritius,  its  main  exports  being 
beans,  acacia  seed,  maize,  salt  fish,  cattle,  goats  and  pigs.  The 
population  is  about  5000,  mostly  settled  around  Port  Mathurin, 
the  only  town  in  the  island.  The  people  are  mainly  French  Creoles, 
with  a  few  Chinese  and  Indians,  and  are  subject  to  the  Government 
of  Mauritius,  which  suppHes  a  Resident  Magistrate.  The  island  is 
a  station  of  the  Eastern  Telegraph  Company,  connecting  to  Cocos- 
Keeling. 

Each  family  usually  cultivates  an  acre  or  several  acres  of  land, 
whereon  they  grow  maize,  sweet  potatoes,  haricot  beans,  pumpkins, 
various  herbs,  onions,  etc.  They  depend,  in  fact,  largely  on  their 
own  plantations  for  food.  At  one  time  a  species  of  mountain-rice, 
which  does  not  require  an  abundance  of  moisture,  was  grown  in 
large  quantities,  but  its  cultivation  was  abandoned  owing  to  the 
depredations  of  small  birds.  Tobacco  grows  well.  Haricot  beans 
are  still  exported.  There  have  lately  been,  however,  only  five  ships 
per  year,  and  these  small  sailing  ships  of  500  tons  down  to  100  tons 
register;  this  makes  it  very  difiicult  to  market  the  produce  of  the 
island.  The  maize  grown  is  barely  enough  for  local  consumption. 


Mr  Snell,  The  Natural  History  of  the  Island  of  Rodrigues     285 

One  of  the  most  profitable  products  of  this  island  is  acacia 
seed,  which  is  exported  to  Mauritius  for  cattle  feeding.  The  acacia 
{Lucaena  glauca),  which  was  introduced  about  seventy  years  ago, 
now  grows  wild  and  flourishes  everywhere,  covering  the  ground 
for  acres,  and  forming  a  dense  almost  impenetrable  scrub,  beneath 
which  nothing  will  grow.  The  cattle  and  goats  are  exceedingly  fond 
of  the  leaves  and  pods,  and  this  is  probably  the  reason  for  its 
spreading  so  extensively,  the  original  plantation  having  been  in  a 
valley  near  Port  Mathurin.  Amongst  other  things  which  have  been 
successfully  grown  may  be  mentioned  coffee,  vanilla,  sugar-cane, 
oranges  and  lemons.  Bananas  and  plantains,  custard  apples, 
strawberries  and  raspberries  are  found  wild.  Many  other  com- 
modities such  as  ginger,  safran  (turmeric)  and  arrowroot  have  also 
been  grown. 

There  is  very  little  real  pasturage  in  Rodrigues,  the  largest 
area  being  in  Malgache  Valley.  Besides  this  there  are  barren  tracts 
round  the  coast  covered  with  coarse  grass,  which  provides  in- 
sufficient subsistence  for  the  stock.  Most  of  the  inhabitants  own 
goats  and  pigs,  on  which  they  rely  for  their  milk  and  meat  supply, 
and  which  are  also  exported.  They  were  allowed  to  run  wild,  but 
measures  have  now  been  introduced  by  the  Government  to  control 
them.    Poultry,  ducks  and  geese  also  thrive  in  the  island. 

Rodrigues  was  originally  covered  with  dense  forests  of  lofty 
trees,  with  corresponding  undergrowth.  Indeed,  according  to 
early  descriptions  its  vegetation  partook  of  the  nature  of  a  regular 
tropical  moist  woodland.  Here  were  to  be  found  flightless  birds, 
the  Solitaires,  and  giant  land  tortoises.  When  Leguat  saw  this 
island  first,  the  scenery  was  such  as  to  call  forth  from  him  such 
designations  as  "a  lovely  isle,"  "an  earthly  paradise."  To-day  its 
grandeur  and  beauty  have  vanished.  There  remains  a  bare  parched 
pile,  on  which  it  is  difficult  if  not  impossible  to  discover  any  corner 
in  its  original  condition.  Many  agencies  are  responsible  for  this 
destruction  and  denudation.  It  has  been  swept  by  fire  many 
times,  accidentally  and  intentionally.  The  goats  devour  the  young 
shoots  and  leaves  of  any  vegetation  within  their  reach.  Pigs  have 
done  their  share,  especially  with  regard  to  the  Latanier  Palm 
(Pandanus),  of  the  nuts  of  which  they  are  very  fond.  Then  there 
are  the  introduced  plants,  which  have  in  many  cases  crowded  out 
the  native  vegetation.  A  notable  example  is  seen  in  the  acacia, 
previously  mentioned,  which  has  spread  into  almost  every  valley 
in  the  island.  A  certain  amount  of  destruction  has  been  done  by 
the  inhabitants,  who  have  cut  timber  over  large  tracts  without 
discrimination.  Though  a  check  has  been  placed  on  this  by  the 
government,  there  still  remains  a  source  of  destruction,  in  that  the 
inhabitants  are  in  the  habit  of  acquiring  year  by  year  fresh  tracts 
of  woodland,  the  undergrowth  of  which  they  cut  down  and  burn, 


286    Mr  Snell,  The  Natural  History  of  the  Island  ofRodrigues 

and  here  they  plant  their  haricot  beans.  They  utiHse  a  tract  of 
land  for  one  season,  and  abandon  it  the  next.  Thus  the  work  of 
destruction  continues.  Many  of  the  older  inhabitants,  at  present 
living  on  the  island,  say  that  they  remember  large  tracts,  which 
are  now  almost  bare  except  for  a  few  Vacoas  (Screw-pines),  being 
originally  covered  with  almost  impenetrable  forest,  but  nobody 
remembers  the  large  expanse  of  coralline  limestone  at  the  south- 
western end  of  the  island  in  any  other  than  its  present  state, 
though  there  are  unmistakeable  traces,  in  roots  and  stumps  em- 
bedded in  the  ground  and  charred  by  fire,  shoAving  that  this  region 
was  also  at  one  time  completely  afforested.  The  large  rifts  are  often 
thirty  feet  or  more  deep,  and  fifteen  to  twenty  yards  wide,  and 
contain  many  fine  old  indigenous  trees  which  have  escaped  destruc- 
tion. The  Valley  of  St  Frangois,  at  the  north-east  end  of  the  island, 
is  perhaps  the  only  other  tract  which  has  escaped  destruction. 

The  commonest  trees  in  the  island  are  the  Vacoas  or  Screw-pines 
(Pandanus),  of  which  there  are  two  species,  both  endemic.  Three 
other  species  have  been  recorded  by  various  authorities,  one  being 
a  native  of  Asia,  and  the  other  two  Madagascar  species.  None  of 
them  occurs  in  Mauritius  or  Reunion,  and  the  evidence  of  their 
occurrence  in  Rodrigues  is  faulty.  There  are  three  species  of 
endemic  palms,  belonging  to  three  genera,  which  are  all  Mascarene. 
Probably  half  the  plants  have  been  destroyed,  but  from  what  is 
left — 297  species  of  Phanerogams,  and  175  species  of  Cryptogams 
(excluding  Marine  Algae) — it  is  clear  that  the  endemic  flora  was 
large  and  of  Mascarene  aSinities.  There  are  only  about  twenty 
species  of  ferns,  the  scarcity  of  this  group  being  accounted  for  by 
the  present  dryness  of  the  island,  in  confirmation  of  which  it  may 
be  remarked  that  the  tree-ferns  of  the  other  Mascarene  islands 
are  not  represented. 

The  present  day  fauna  is  not  large.  The  extinct  fauna  has  proved 
to  be  of  very  great  interest,  particularly  in  the  case  of  the  Solitaire 
(Pezophaps  solitaria,  Gmel.),  the  extinct  Didine  bird  related  to  the 
i)odo  of  Mauritius.  Considerable  collections  of  the  remains  of  this 
bird  have  been  made  from  the  limestone  caves,  where  also  the 
remains  of  other  extinct  birds  and  of  the  giant  Land  Tortoise  have 
been  found.  Our  main  knowledge  of  the  recent  fauna  is  due  to  the 
labours  of  the  naturalists  attached  to  the  Transit  of  Venus  Ex- 
peditions carried  out  in  1874-5. 

The  marine  fauna  is  in  general  of  the  Indo-Pacific  type. 

The  only  indigenous  mammal  found  in  the  island  is  a  fruit-bat, 
Pteropus  rodericensis,  Dobson,  which  is  peculiar  to  Rodrigues.  The 
introduced  mammals,  other  than  those  already  mentioned,  are 
deer,  rabbits,  rats,  mice  and  cats,  the  latter  being  left  by  the 
Dutch  to  destroy  the  rats. 

Sir  Edward  Newton,  K.C.M.G.,  published  a  list  of  Rodrigues 


Mr  Snell,  The  Natural  History  of  the  Island  of  Rodrigues     287 

birds  in  his  "List  of  the  Birds  of  the  Mascarene  Islands"  {Trans. 
Norfolk  and  Nonvich  Naturalists'  Society,  vol.  iv,  President's 
Address). 

The  Fresh  Water  Fishes,  as  far  as  known,  belong  to  species 
which  inhabit  the  fresh  waters  of  the  Mascarene  Islands  generally, 
with  the  exception  of  two  Grey  Mullets,  which  were  collected  by 
the  Transit  of  Venus  Expedition,  and  were  described  as  new. 

Further  collections  in  certain  groups  have  recently  been  made 
by  Mr  H.  P.  Thomasset  and  Mr  H.  J.  Snell,  who  visited  the  island 
during  the  period  August  to  November,  1918,  with  a  view  to  im- 
proving our  knowledge  of  the  insect  fauna. 

Mr  Snell  visited  practically  every  part  of  the  island,  with  the 
exception  of  the  valley  of  St  Francois,  and  a  small  district  round 
the  Riviere  Coco.  The  best  collecting  ground  he  found  to  be  un- 
doubtedly the  Grande  Riviere  Valley,  which  he  worked  right  up  to 
Mount  Limon.  The  islands  on  the  reef  were  also  visited,  but  con- 
tained very  little  of  interest,  as  they  have  been  burnt  over  in  recent 
years,  and  are  now  covered  with  rough  coarse  grass  and  short 
scrub  {Tournefortia,  Pemphis,  etc.).  These  islands,  particularly 
Gombranil  and  Flat,  were  formerly  nesting  places  for  sea-birds, 
which  seem  to  have  disappeared,  only  a  few  white  terns  and 
boobies  being  found  on  Sandy  and  Coco  Islands,  which  were  some 
years  ago  planted  with  firs. 

In  the  deepest  ravines  were  commonly  seen  the  fruit-bats  or 
flying-foxes,  feeding  on  the  flower  of  a  kind  of  aloe,  of  which  they 
seem  very  fond,  and  also  on  wild  figs,  mangoes,  etc.  Geckos  were 
abundant  in  warm  and  sheltered  spots,  particularly  in  all  habita- 
tions. Their  eggs  were  frequently  found  in  nests  (usually  composed 
of  dry  Sow-thistle  bloom)  under  rocks  and  in  crevices.  Two  species 
only  have  been  recorded:  Gehyra  mutilata,  Gray,  and  Phelsuma 
cepedianum,  Gray;  the  latter  is  common  in  Madagascar,  Mauritius 
and  Reunion,  but  is  rare  in  Rodrigues.  Freshwater  fishes  were 
found  in  many  of  the  streams,  in  which  also  eels  were  quite 
common. 

There  are  in  the  island  a  Land  Planarian,  Geojplana  whartoni, 
Gull.,  and  a  Land  Nemertean,  Tetrastemma  rodericanum,  Gull. 
Both  are  peculiar  to  Rodrigues,  but  the  former  has  not  been  ade- 
quately described.  (Mr  Thomasset  subsequently  obtained  a  Land 
Planarian  from  Mauritius,  a  new  locality  for  these.)  They  were 
found  under  decaying  logs,  sometimes  on  the  bark,  under  the 
bark,  or  in  the  wood;  the  Nemertean  appeared  to  exist  in  far 
greater  quantities  than  the  Land  Planarians,  but  they  often  live 
together  in  the  same  situation.  Earthworms  were  not  abundant. 
Amongst  the  Crustacea  collected,  large  numbers  of  an  Amphipod 
were  found  under  stones,  dead  leaves,  etc.,  wherever  the  ground 
was  moist.  In  all  the  streams  were  to  be  found  freshwater  shrimps 

VOL.  XIX.  PART  VI.  20 


288    Mr  Snell,  The  Natural  History  of  the  Island  of  Rodrigues 

and  a  crayfish.  Woodlice  were  abundant  in  deca}dng  vegetable 
matter,  the  largest  specimens  being  obtained  from  rotting  banana 
stems. 

Myriapoda  were  common  throughout  the  island.  Large  centi- 
pedes live  on  the  corals  on  the  west  side  of  the  island,  attaining 
sometimes  a  length  of  twelve  inches.  Hardly  a  lump  of  debris  can 
be  turned  over  without  disclosing  one  or  more  of  these  creatures. 
The  Transit  of  Venus  Expedition  obtained  twelve  species  of 
Myriapods,  of  which  eleven  were  new.  There  is  a  single  species  of 
scorpion,  Tityus  marmoreus,  Koch,  and  in  addition  the  Transit  of 
Venus  Expedition  obtained  twenty-seven  species  of  Arachnida, 
eleven  being  new ;  unfortunately  Mr  Snell  could  not  obtain  a  supply 
of  alcohol  adequate  to  preserve  these. 

In  the  Insect  collections  among  the  Orthoptera,  the  Forficulidae 
are  represented  by  eleven  specimens,  probably  Anisolabis  varicornis, 
Smith.  Of  the  Blattidae,  Periplaneta  americana,  Linn,  and  Leu- 
cophaea  surinamensis,  Fab.  are  among  the  five  species  previously 
recorded,  whilst  there  are  two  other  species  in  Mr  Snell's  collection 
at  present  undetermined.  One  species  of  Mantidae  occurs  in  the 
island,  viz.  Polyspilota  aeruginosa,  Goeze,  of  wide  distribution.  Of 
the  Gryllidae  there  are  three  species  in  the  present  collection: 
Acheta  bimaculata,  de  Geer,  found  also  in  Africa  and  S.  Europe; 
Curtilla  africana,  Beauv.,  found  also  in  Africa,  Asia,  Australia,  and 
New  Zealand  (introd.?);  and  a  species  of  Ornebius  near  syrticus, 
Bolivar,  but  larger  and  more  brightly  coloured  than  the  Seychelles 
specimens  of  this  species.  Besides  the  first  of  these,  the  Transit  of 
Venus  Expedition  obtained  three  other  species.  Among  the 
Phasgonuridae  we  have  Conocephaloides  differens,  Serv.  and 
Anisoptera  iris,  Serv.,  both  previously  recorded  by  the  Transit  of 
Venus  Expedition.  In  addition  the  present  collection  contains  a 
specimen  of  apparently  another  species  of  Anisoptera,  resembhng 
A.  conocephala,  Linn.,  which  occurs  in  Spain,  Africa,  and  the 
Seychelles.  There  are  two  species  of  Locustidae:  Locusta  danica, 
Linn.,  a  cosmopolitan  species,  and  Chortoicetes  rodericensis,  Butl., 
described  from  Rodrigues,  and  not  found  elsewhere. 

The  Neuroptera  comprise  a  few  specimens  of  a  Termite,  and 
specimens  of  one  species  of  Hemerobiidae  and  of  one  species  of 
Chrysopidae.  It  may  here  be  mentioned  that  Dr  H.  Scott  found  a 
species  of  Termite  working  in  the  wood  at  the  bottom  of  a  fighter 
in  Victoria  harbour,  Mahe,  Seychelles.  This  indicates  a  possible 
explanation  of  the  existence  of  Termites  in  such  a  locafity  as 
Rodrigues,  where  any  indigenous  Termites  would  probably  be 
exterminated  by  the  fires  which  have  repeatedly  devastated  the 
island.  Until  the  Termites  in  Mr  Snell's  collection  have  been 
identified,  no  statement  of  course  can  be  ventured  regarding  the 
distribution  of  this  species.   Mr  Gulfiver,  on  the  Transit  of  Venus 


Mr  Snell,  The  Natural  History  of  the  Island  of  Rodrigues    289 

Expedition,  secured  one  specimen  of  Myrmeleon  obscurus,  Rambur. 
This  species  was  described  from  Mauritius,  and  is  widely  distributed 
in  Africa. 

The  Odonata  consist  of  six  species,  as  follows : 

Pantala  flavescens.  Fab.,  occurs  in  all  the  warmer  parts  of  the 
world,  but  not  in  Europe. 

Tramea  limhata,  Desj.,  a  very  variable  species  of  wide  dis- 
tribution, described  from  Mauritius. 

Orthetrum  hrachiale,  P.  de  Beauv.  Found  elsewhere  in  Zanzibar, 
Congo,  etc. 

Anax  imperator  mauricianus,  Rambur.  Agrees  with  a  specimen 
in  the  Museum  of  Zoology,  Cambridge,  named  by  Campion.  The 
species  was  also  taken  by  Gulliver,  on  the  Transit  of  Venus  Ex- 
pedition. 

Ischnura  senegalensis,  Rambur.  Widely  distributed  in  tropical 
Asia  and  Africa. 

Agrion  ferrugineum,  Rambur.  One  specimen  was  taken  by 
GulHver.  The  present  collection  contains  several  specimens. 

The  collection  of  Hymenoptera,  exclusive  of  Ants,  contains  two 
species  of  Tubulifera,  eleven  species  of  Aculeata,  and  approxi- 
mately 170  specimens  (of  about  twenty  species)  of  Parasitica.  The 
two  species  of  TubuUfera,  for  the  identification  of  which  I  am 
indebted  to  Mr  F.  D.  Morice  of  the  British  Museum  of  Natural 
History,  are  Chrysis  {Pentachrysis)  lusca,  Fab.,  found  also  in  India, 
Ceylon  and  Mauritius,  and  Philoctetes  coriaceus,  Dahlb.,  known 
also  from  East  and  South  Africa.  Of  the  Aculeata,  the  Formicidae 
are  not  yet  determined,  and  a  species  of  Halictus  is  at  present 
unidentified.  The  remainder  of  the  Aculeates  are  as  follows : 

Megachile  disjuncta,  Fab.  Common  in  India;  recorded  also 
from  Mauritius.  (M.  lanata,  Fab.,  is  recorded  by  Smith  as  having 
been  taken  by  Gulhver  on  the  Transit  of  Venus  Expedition.) 

Megachile  rufiventris,  Guer.  Found  elsewhere  in  East  and  South 
Africa,  Mauritius  and  Seychelles;  previously  taken  in  Rodrigues 
by  GulUver. 

Apis  unicolor,  Latr.  Previously  taken  in  Rodrigues  by  Gulhver. 
Found  in  the  Seychelles,  Amirantes,  Chagos  (Diego  Garcia,  Peros 
Banhos).   Commoner  in  Madagascar. 

Odynerus  trilobns,  Fab.  This  species  has  not  been  previously 
recorded  from  Rodrigues.  It  is  common  and  widely  distributed, 
being  known  from  Madagascar,  Mauritius,  Reunion  and  South 
Africa. 

Polistes  macaensis,  Fab.  Previously  taken  by  Gulhver  and 
listed  as  P.  hebraeus,  Linn,  There  seems  to  have  been  considerable 
confusion  over  these  names,  as  Cameron  {Trans.  Linn.  Soc.  (2), 
vol.  XII,  p.  71)  hsts  this  species  as  P.  hebraeus,  Fab.,  stating  that 
it  is  known  from  Rodrigues.    Dr  R,  C.  L.  Perkins  has,  however, 

20—2 


290    Mr  Snell,  The  Natural  History  of  the  Island  of  Rodrigues 

demonstrated  the  differences  between  the  male  P.  macaensis  and 
male  P.  hebraeus.  (See  Ent.  Mo.  Mag.  (2),  vol.  xii,  1901,  p.  264.) 
P.  macaensis  is  known  also  from  Seychelles,  Amirantes,  Chagos 
(Salomon  Islands,  Diego  Garcia),  and  Mauritius. 

Scolia  (Dielis)  grandidieri,  Sauss.  I  am  indebted  to  Mr  Rowland 
E.  Turner  of  the  British  Museum  of  Natural  History  for  the 
identification  of  this  species.  He  states  that  the  specimens  under 
review  are  of  "a  form  of  D.  grandidieri,  Sauss.  from  Madagascar, 
with  a  few  more  punctures  on  the  abdomen  than  in  that 
species." 

Ampulex  compressa,  Fab.,  not  previously  recorded  from 
Rodrigues.  Common  from  Eastern  Europe  to  China,  and  also  in 
Africa. 

Passaloecus  (Polemistus)  macilentus,  Sauss.  Mr  R.  E.  Turner  has 
kindly  identified  this  species  for  me.  He  states  (in  litt.)  that  "Mr 
Morice  considers  that  Philoctetes  coriaceus,  Dahlb.  is  probably 
parasitic  on  this,  as  species  of  Passaloecus  are  often  attacked  by 
small  Chrysids."  The  species  was  described  from  Madagascar. 

Sceliphron  hengalense,  Dahlb.  (  =  Peolpaeus  convexus,  Sm.). 
Mr  Turner  has  confirmed  my  identification  of  this  species.  He 
adds:  "This  is  probably  an  imported  species,  as  species  of  the 
genus  build  mud  nests  on  ships  and  are  carried  in  that  way  from 
place  to  place." 

Trypoxylon  errans,  Sauss.  Not  previously  recorded  from 
Rodrigues.  Found  also  in  Mauritius  and  the  Seychelles. 

There  are  approximately  750  specimens  of  Coleoptera,  of  pos- 
sibly 100  species;  640  specimens  of  Diptera,  of  at  least  seventy 
species;  and  360  specimens  of  Hemiptera,  of  some  forty-five 
species.  These  have  not  yet  been  critically  examined. 

In  the  Lepidoptera,  seven  species  of  Butterflies  were  collected 
by  Mr  Grulliver  on  the  Transit  of  Venus  Expedition.  Of  these  one 
species  is  not  represented  in  Mr  Snell's  collection,  viz.  Hesperia 
forestan,  Cr.    The  list  of  Butterflies  is  as  follows : 

'^Melanitis  leda,  Linn.  '\*Zizera  lysimon,  Hiibn, 

*Danais  chrysippus,  Linn.  "f^Polyommatus  boeticus,  Linn. 

Precis  rhadama,  Boisd.  *Tarucus  telicanus,  Lang. 

*Hypolimnas  misippus,  Linn.  Parnara  borbonica,  Boisd. 
"f^Atella  phalantha,  Drury 

Among  the  Moths  (Heterocera),  exclusive  of  the  Pyralidae, 
Tortricidae,  and  Tineidae,  though  Gulliver's  collection  contained 
only  twelve  species,  five  of  these  were  species  not  represented  in 
Mr  Snell's  collection,  Mr  Snell  obtained  three  species  of  Sphingidae, 

*  Of  wide  distribution. 

■]•  Not  previously  recorded  from  Rodrigues. 


Mr  Snell,  The  Natural  History  of  the  Island  of  Rodrigues    291 

one  species  of  Arctiidae,  twenty-five  species  of  Noctuidae,  and 
two  species  of  Geometridae,  as  follows : 

^Acherontia  atropos,  Linn.  "f^Erias  insulana,  Boisd. 
'f*Herse  convolvuli,  Linn.  *An,ua  tirhaca,  Cr. 

"I"  Hippotion  aurora,  Roth.  &  Jord.    Achaea  trapezoides,  Guen. 
i*Utetheisa  pulchelloides,  Hamps.     Achaea  finita,  Guen. 
'f*Chloridea  obsoleta,  Fab.  *Parallelia  algira,  Linn. 

"f^Agrotis  ypsilon,  Linn.  *Chalciope  hyppasia,  Cr. 

■f*Cirphis  loreyi,  Dup.  "f^Mocis  undata',  Fab, 
■j"  Cirphis  leucosticha,  Hamps.         *Phytometra  chalcytes,  Esp. 

(  =  insulicola,  Saalm.)  *Cosmophila  erosa,  Hiibn. 

"f^Perigea  capensis,  Guen.  "f^Dragana  pansalis,  Walk. 

^^Eriopus  maillardi,  Guen.  ^*Magulaha  imparata,  Walk. 

*Prodenia  litura,  Fab.  ^^Hydrillodes  lentalis,  Guen. 

*Spodoptera  abyssinia,  Guen.  "f^Hypena  masurialis,  Guen. 

Athetis  expolita,  Butl.  ^*Hyblaea  puera,  Cr. 

"fEublemma  apicimacula,  Mab.  f*Craspedia  minorata,  Boisd. 

*Amyna  octo,  Guen.  ^*Thalassodes  quadraria,  Guen. 

The  five  species  collected  by  Mr  Gulliver  and  not  represented 
in  the  present  collection  are  as  follows: 

*Argina  cribraria,  Clerck.  (Hypsidae). 

*Nodaria  externalis.  Walk,  (redescribed  as  Diomea  bryophiloides, 
Butl.)  (Noctuidae). 

Pericyma  turbida,  Butl.  (Noctuidae).   Peculiar  to  Rodrigues. 
*Achaea  catella,  Guen.  (Noctuidae). 
*Mocis  repanda,  Fabr.  (Noctuidae). 

Butler  listed  a  species  as  Laphygma  cycloides,  Guen.,  apparently 
in  error,  as  Sir  George  Hampson  has  in  his  Catalogue  placed  the 
record  under  Spodoptera  abyssinia,  Guen. 

There  are  about  180  specimens,  of  some  thirty  species,  of 
Micro-lepidoptera.  These  have  not  yet  been  worked  out. 

The  collections  made  by  Mr  Snell  are  of  importance  as  showing 
more  definitely  the  relations  of  Rodrigues  with  the  other  islands 
in  the  vicinity.  Undoubtedly  the  fauna  has,  with  the  flora,  suffered 
considerably  from  the  devastating  effects  of  the  fires  which  have 
so  frequently  swept  the  island,  but  investigation  of  the  collections 
of  the  groups  not  yet  worked  out,  will  undoubtedly  show  that  con- 
siderable traces  of  the  indigenous  fauna  still  exist,  and  will  serve 
to  indicate  with  greater  accuracy  the  affinities  of  Rodrigues  with 
the  neighbouring  islands. 

*.  Of  wide  distribution. 

f  Not  previously  recorded  from  Rodrigues. 


292     Mr  Snell,  The  Natural  History  of  the  Island  of  Rodrigues 

Bibliography. 

Legtjat,   Franqois.     Voyages  et  aventures  en  deux  lies  Desertes  des  Indes 

Orientales  (1650-1698). 
Grant,  C.  History  of  Mauritius  and  the  neighbouring  Islands  (1801). 
(Pridham,  C?)   An  Account  of  the  Island  of  Mauritius  and  its  Dependencies 

(1842). 
Strickland  and  Melville.   The  Dodo  and  its  Kindred  (1848). 
HiGGiN,  E.    "Remarks  on  the  Country,  Products  and  Appearance  of  the 

Island  of  Rodrigues,  with  opinions  as  to  its  future  Colonization."   Joum. 

Boy.  Geogr.  Soc.  xix,  Pt  I,  1849,  p.  17. 
Newton,  E.   "Notes  of  a  Visit  to  the  Island  of  Rodrigues."  Ibis,  vol.  i,  (new 

series),  1865. 
Balfour,  I.  B.  and  others.   "An  Account  of  the  Petrological,  Botanical  and 

Zoological  Collections  made  in  Rodrigues  during  the  Transit  of  Venus 

Expeditions  in  1874-5."  Phil.  Trans.  Boy.  Soc.  vol.  CLXvm  (extra  volume), 

1878. 
Oliver,  S.  P.  (Edit.)  The  Voyage  of  Frangois  Leguat  of  Bresse  (2  vols.  1891). 

(Transcribed  from  the  Original  English  Edition  for  the  Hakluyt  Society.) 
Gardiner,  J.  S.    "Islands  in  the  Indian  Ocean,  Mauritius,  Seychelles,  and 

Dependencies,   1914."    In  the  volume  on  Africa,  Oxford  Survey  of  the 

British  Empire. 
Newton,  Sir  E.,  K.C.M.G.    "The  Birds  of  the  Mascarene  Islands."    Presi- 
dential Address.   Trans.  Norfolk  and  Norwich  Naturalists'  Society,  vol.  iv. 
Encyclopaedia  Britannica. 


Miss  Haviland,  Note  on  the  Life  History  of  Lygocerus      293 


Preliminary  Note  on  the  Life  History  of  Lygocerus  {Procto- 
trypidae),  hyperparasite  of  Aphidius.  By  Maud  D.  Haviland, 
Fellow  of  Newnham  College.  (Communicated  by  Mr  H.  H. 
Brindley.) 

[Read  10  November  1919.] 

Plant  lice  are  frequently  parasitized  by  certain  Braconidae  of 
the  family  Aphidiidae.  The  parasite  oviposits  in  the  haemocoele 
of  the  aphis,  and  the  larva,  during  development,  consumes  the 
viscera  of  the  host.  At  metamorphosis  nothing  remains  but  the 
dry  skin,  within  which  the  Aphidius  spins  a  cocoon  for  pupation. 

At  this  stage,  the  Aphidius  itself  is  liable  to  be  parasitized  in 
turn  by  certain  Cynipidae,  Chalcidae,  and  Proctotrypidae.  The 
two  former  are  known  to  be  hyperparasites,  but  the  Proctotry- 
pidae have  hitherto  been  considered  doubtful,  although  some 
writers  have  suspected  that  they  are  hyperparasites  of  the  Aphidius, 
and  not  parasites  of  the  aphis.  Gatenby  in  his  paper:  "  Notes  on  the 
Bionomics,  Embryology,  and  Anatomy  of  certain  Hymenoptera 
Parasitica"  (Journ.  Linn.  Soc.  1919,  vol.  xxx,  pp.  387-416)  says: 
". .  .1  am  inclined  to  support  the  view  that  the  Proctotrypid  is  a 
parasite,  and  not  a  hyperparasite." 

The  following  is  a  summary  of  some  observations  made  in  the 
summer  of  1919,  on  two  Proctotrypids  of  the  genus  Lygocerus. 
I  am  much  indebted  to  Professor  Kieffer,  who  has  kindly  identified 
them  for  me  as  L.  testaceimanus,  Kieff.,  hyperparasite  of  Aphidius 
salicis,  Hal.,  parasite  of  Aphis  saliceti,  Kalt.,  from  the  willow;  and 
L.  cameroni,  Kieff.,  hyperparasite  of  Aphidius  ervi,  Hal.,  parasite 
of  Macrosiphum.  urticae  from  the  nettle.  The  following  notes 
probably  apply  to  both  species,  but  the  observations  were  made 
more  especially  upon  the  latter.  It  was  found  also  that  in  cap- 
tivity L.  testaceimanus  would  oviposit  on  Aphidius  ervi.  The 
Proctotrypids  do  not  confine  their  attacks  to  the  Aphidiidae,  but 
their  larvae  may  also  be  found  feeding  on  the  larvae  of  other 
Chalcid  or  Cynipid  hyperparasites  of  that  family ;  and  indeed  once 
or  twice  were  observed  upon  dead  pupae  of  their  own  species.  One 
remarkable  instance  of  hyperparasitism  came  under  notice.  An 
aphis  {Macrosiphum  urticae)  was  parasitized  by  an  Aphidius  (A. 
ervi).  The  latter  had  been  hyperparasitized  by  a  Chalcid,  of  species 
unknown,  which  immediately  after  pupation  had  been  attacked 
by  another  hyperparasite,  either  Chalcid  or  Cynipid,  whose  identity 
is  not  yet  determined.  This  second  hyperparasite  in  turn  had  been 
attacked  by  Lygocerus  cameroni,  and  the  larva  was  in  the  second 
instar  when  the  cocoon  was  opened.  We  may  ask,  where  are  the 
limits  to  this  hyperparasitism? 


294     Miss  Haviland,  Note  on  the  Life  History  of  Lygocerus 

Lygocerus  cameroni  was  fairly  common  round  Cambridge  in 
1919,  from  mid-July  to  the  end  of  August.  The  female  selects  an 
aphis-cocoon  containing  a  full-grown  larva  or  newly  transformed 
pupa  of  Aphidius,  and  runs  round  it  with  much  excitement, 
tapping  it  with  her  antennae.  Oviposition  takes  from  30-60 
seconds,  the  insect  meanwhile  standing  either  on  the  top  of  the 
cocoon  facing  the  anterior  end,  or  on  the  leaf  behind,  with  her  back 
to  it.  Either  way,  the  ovipositor  is  brought  into  the  angle  of  the 
host's  body,  as  it  lies  curled  inside.  Sometimes  two  or  three  eggs, 
the  result  of  successive  ovipositions  by  different  females,  are 
found  on  the  same  host. 

The  egg,  which  is  laid  on  the  upper  surface  of  the  abdomen  of 
the  Aphidius,  measures  -25  x  -10  mm.  It  is  translucent,  white, 
and  elliptical,  with  marked  longitudinal  striae  of  the  chorion,  and 
a  minute  stalk  at  one  end.  Treatment  of  the  egg  with  lacto-phenol 
and  cotton-blue  showed  the  presence  of  bodies  resembling  the 
symbiotes  from  the  pseudovitellus  of  Aphides.  The  egg  hatches  in 
about  twenty  hours. 

The  larva  of  the  first  instar  is  a  maggot  shaped  form,  with 
thirteen  body  segments  and  a  head  furnished  with  two  minute 
papillae.  The  mouth,  which  is  circular  and  very  small,  contains 
two  simple  chitinous  mandibles  set  well  behind  the  hood-hke 
labrum  and  the  labium.  The  mid-gut,  which  at  this  stage  does  not 
communicate  with  the  rectum,  is  large  and  globose,  and  its  con- 
tents tinge  the  transparent  body  pale  yellow.  Later  on,  when  the 
host  dies,  they  become  brown.  The  tracheal  system  consists  of  two 
lateral  longitudinal  trunks,  united  by  an  anterior  and  posterior 
commissure.  When  newly  hatched,  there  are  two  open  spiracles 
between  the  first  and  second  and  on  the  fourth  segments,  but 
soon  afterwards  the  spiracles  of  the  third  and  fifth  segments 
become  functional.  The  larva  is  active  and  crawls  over  the  host's 
body.  This  instar  lasts  from  twenty  to  twenty-four  hours,  and  the 
dimensions  are  about  -45  x  -22  mm. 

The  larva  of  the  second  instar  differs  from  that  of  the  first 
chiefly  in  the  size,  which  is  -70  x  -35  mm.,  and  in  the  tracheal 
system.  The  ramifications  of  the  latter  are  more  numerous,  the 
dorso- ventral  branches  of  the  second  segment  become  visible,  and 
the  spiracular  trunks  of  segments  six,  seven,  and  eight  appear, 
though  their  spiracles  are  not  open.  The  duration  of  this  instar  is 
about  thirty-six  hours,  and  at  this  time  the  host  usually  dies,  and 
its  body  becomes  blackened  and  shrunken. 

In  the  third  instar,  the  papillae  on  the  head  disappear,  the  body 
becomes  more  globose,  and  the  greater  proportionate  development 
of  the  three  first  segments  causes  the  head  to  be  bent  round  to  the 
ventral  side.  The  dimensions  are  about  1-00  x  -75  mm.  The  spiracles 
of  the  sixth,  seventh  and  eighth  segments  open,  and  the  spiracular 


{Prodotrypidae),  hyperparasite  of  Aphidius  295 

trunk  of  the  second  segment  becomes  visible.  In  addition,  two 
short  spiracular  trunks  can  be  made  out  on  the  ninth  and  tenth 
segments;  but  these  never  become  functional,  and  they  disappear 
in  the  later  stages  of  development.  This  instar  lasts  from  about 
thirty-five  to  forty  hours. 

In  the  fourth  instar,  which  lasts  about  two  days,  the  Procto- 
trypid  grows  rapidly,  and  when  mature  measures  1-67  x  -83  mm. 
The  remainder  of  the  host  is  quickly  consumed,  and,  just  before 
metamorphosis,  the  mid-gut  opens  into  the  rectum,  and  its  con- 
tents are  voided  into  the  cocoon.  The  larva  is  active  and  wriggles 
about  freely  inside  the  aphis  skin,  aided  possibly  by  a  curious 
caudal  appendage;  and  by  these  movements  the  faeces,  together 
with  the  host's  skin,  are  kneaded  into  a  moist  compact  pellet  on 
the  ventral  side  of  the  body. 

The  full  grown  larva  is  yellowish  white,  and  each  segment  has 
a  double  row  of  short  chitinous  spines.  The  thorax  is  large  and 
broad,  while  the  abdominal  segments  taper  away  somewhat  to  the 
eleventh,  which  bears  a  short  stout  appendage  furnished  with 
spines.  The  head  is  turned  completely  under  the  thorax,  and  the 
tracheal  system  does  not  differ  essentially  from  that  of  the  pre- 
ceding instar.  No  larval  antennae  nor  maxillary  nor  labial  palpi 
seem  to  exist  at  this  stage. 

Lygocerus  does  not  produce  silk,  but  pupates  in  the  cocoon  made 
previously  by  the  Aphidius  inside  the  skin  of  the  aphis.  The  period 
of  pupation  is  fourteen  to  sixteen  days.  When  ready  to  emerge, 
the  imago  gnaws  a  hole  somewhere  on  the  upper  side  of  the  cocoon, 
and  creeps  out.  So  far,  no  parthenogenetic  ovipositions  have  been 
observed,  and  two  broods,  certainly,  and  possibly  more,  may  occur 
in  the  season.  The  life  of  the  imagoes  is  generally  five  or  six  days, 
but  they  may  live  as  many  as  ten.  Examples  in  captivity  were 
observed  to  feed  on  the  sap  oozing  from  cut  leaves,  and  on  honey- 
dew  dropped  by  the  aphides,  but  they  seemed  to  live  as  long  and  to 
remain  as  vigorous  when  no  food  was  supplied. 


296  Mr  Warburton,  Note  on  the  solitary  wasj), 


Note  on  the  solitary  wasp,  Crabro  cephalotes.  By  Cecil 
Warbueton,  M.A.,  Christ's  College. 

[Read  10  November  1919.] 

Last  summer  a  small  colony  of  C.  cephalotes  took  possession  in 
my  garden  of  a  log  of  elmwood  which  was  kept  as  an  example  of 
a  woodpecker's  nest.  The  entrance  hole  of  the  woodpecker  was 
there,  and  just  below  it  the  log  had  been  sawn  through  so  that  the 
internal  cavity  could  be  examined. 

The  first  advent  of  the  wasps  was  not  noticed,  but  in  the  first 
week  of  August  a  wasp  was  observed  entering  the  hole,  and  this 
led  to  an  investigation  of  the  log,  which  presented  signs  of  boring 
in  the  half-decayed  heart-wood.  One  of  the  wasps  had  attacked 
the  log  from  the  top  and  its  operations  could  be  noted  with  more 
or  less  exactness,  but  the  others  passed  in  and  out  by  the  wood- 
pecker's hole,  and  it  was  impossible  to  recognise  individuals  or  to 
follow  their  work  without  constantly  disturbing  it  by  opening  up 
the  log,  with  the  risk  of  inaccurately  replacing  the  two  halves. 
The  log  was  nevertheless  opened  several  times  during  the  first  half 
of  August,  but  it  was  then  thought  better  to  let  the  wasps  finish 
their  work  without  further  disturbance. 

That  the  wasps  are  not  easily  diverted  from  their  labours  the 
following  facts  sufficiently  demonstrate.  The  log  was  moved  several 
yards,  to  a  spot  more  convenient  for  observation.  The  wasp 
working  on  the  top  (hereafter  referred  to  as  wasp  No.  1)  was 
captured  in  a  glass  tube  and  examined  for  identification,  but  on 
being  liberated  continued  working  as  before.  Close  observation, 
with  a  hand  lens,  did  not  deter  this  wasp  from  entering  its  burrow 
without  hesitation  in  the  course  of  its  operations,  nor  were  the 
other  wasps  disconcerted  by  the  removal  of  the  lid  on  several 
occasions  at  an  early  stage  of  their  work.  As  a  rule  no  attention 
was  paid  to  anyone  sitting  silently  near  the  log,  but  it  must  be 
recorded  that  on  one  occasion  a  wasp  returning  with  a  fly  appar- 
ently objected  to  the  dress — light  with  dark  spots — of  a  lady  sitting 
near  at  hand,  and  after  a  close  investigation  from  many  points  of 
view,  retired  instead  of  entering  the  log.  To  ascertain  if  wasp  No.  1 
were  at  home  or  not  I  was  in  the  habit  of  placing  a  stout  straw  in 
its  burrow — protruding  an  inch  or  more.  One  would  have  thought 
that  on  returning  home  and  finding  such  an  object  impeding  its 
entrance  the  insect  would  manifest  some  perturbation  and  either 
refuse  to  enter  or  take  some  measures  to  remove  the  obstacle.  It 
did  nothing  of  the  kind,  but  absolutely  disregarded  the  straw, 
pushing  past  it  even  when  laden  with  a  fly.   It  was  several  times 


Crabro  cephalotes  297 

ejected  together  with  the  frass  from  new  tunnelling  operations, 
but  never  otherwise. 

Continuous  observation  of  work  that  went  on  for  many  hours 
a  day  for  about  three  weeks  was,  of  course,  impossible,  but  on 
several  days,  especially  during  the  week  Aug.  18 — 25,  operations 
were  watched  for  spells  of  an  hour  or  two  at  a  time,  and  the  exact 
times  of  ingress  and  egress  carefully  noted.  The  notes  which 
immediately  follow  especially  concern  wasp  No.  1. 

The  hole  was  sometimes  clear,  sometimes  choked  with  "saw- 
dust." After  watching  for  a  time  the  "sawdust"  would  be  seen  to 
heave  up  and  form  a  mound  over  the  hole.  Then  the  wasp  would 
emerge  and  proceed  to  remove  the  frass,  butting  it  away  from  the 
neighbourhood  of  the  hole  with  its  head.  Sometimes  in  the  course 
of  its  excavations  the  wasp  would  emerge,  fly  away  for  a  time,  and 
return  empty  handed  to  resume  its  digging. 

On  Aug.  19  it  was  seen  to  be  carrying  home  flies,  and  the  per- 
formance was  watched  for  an  hour,  and  the  following  times  were 
noted: 

Returned  with  fly,  9.37,  9.48,  10.18,  10.31. 
Emerged,  9.40,  9.55,  10.25,  10.39. 

Thus  four  flies  were  caught  in  the  hour,  and  the  times  spent  in 
capturing  three  of  them  were  8',  23'  and  6'  respectively,  while 
3',  7',  7'  and  8'  were  occupied  in  packing  the  four  flies  into  the 
burrows.  To  find,  capture,  paralyse  and  bring  home  the  right  kind 
of  fly  in  six  minutes  strikes  one  as  a  remarkable  feat.  From  further 
observations  it  appeared  that  the  operation  usually  occupied  about 
a  quarter  of  an  hour.  None  but  "hover  flies"  (Syrphidae)  were 
taken  by  any  of  the  wasps,  and  the  prey  was  generally  Syrphus 
halteatus,  a  species  almost  as  large  as  the  wasp  itself.  It  was, 
nevertheless,  carried  with  perfect  ease,  arranged  longitudinally, 
head  foremost  beneath  its  captor,  and,  I  believe,  venter  to  venter. 
No  preliminary  examination  of  the  hole  was  ever  made  before 
carrying  the  fly  in,  such  as  Fabre  has  recorded  in  the  case  of  some 
wasps.  About  noon  on  Aug.  21  this  wasp  apparently  ceased 
workino;.  There  were  no  signs  of  activitv  that  afternoon  nor  the 
following  morning. 

On  Aug.  22  about  3  p.m.  a  wasp  (wasp  No.  2)  was  seen  to  come 
out  of  the  woodpecker's  hole  and  alight  on  the  top  of  the  log, 
which  it  proceeded  to  explore.  It  found  No.  I's  burrow  and 
entered  it  for  a  short  distance,  after  which  it  flew  away.  Nothing 
further  was  noted  till  the  evening  of  Aug.  23,  when  on  returning 
home  at  5.30  I  noticed  a  heap  of  frass  on  the  top  of  the  hole.  At 
6.20  a  wasp  arrived  and  after  pointing  at  the  main  entrance, 
seemed  to  change  its  mind  and  alighting  on  the  top,  entered  No. 
I's  hole.   Its  behaviour  convinced  me  that  it  was  not  No.  1,  but  it 


298  Mr  Warburton,  Note  on  the  solitary  wasp, 

might  very  well  be  wasp  No.  2.  Anyhow  it  entered  the  burrow, 
and  by  7.50  it  had  turned  out  more  "sawdust"  containing  several 
of  the  flies  so  carefully  stored  up  by  wasp  No.  1 !  The  explanation 
that  first  occurred  to  one  was  that  the  wasp  wanted  to  dig,  and 
naturally  found  it  easier  to  work  w^here  someone  had  been  before. 
Such  a  defective  instinct  would,  however,  militate  against  the 
preservation  of  the  race.  Moreover  there  were  no  further  develop- 
ments, and  No.  2  remained  satisfied  with  undoing  some  of  No.  I's 
work.  A  wild  suggestion  did  occur  to  me,  which  I  will  give  for 
what  it  is  worth.  Is  it  possible  that  one  of  those  working  from  the 
interior  became  aware  of  operations  from  the  outside  which  might 
imperil  the  results  of  its  own  labours,  and  proceeded  to  put  a 
stop  to  them? 

With  regard  to  the  remaining  wasps,  which  entered  by  the 
woodpecker's  hole  and  worked  from  the  inside,  the  following  notes 
may  be  given. 

The  earlier  hasty  inspections  of  the  interior  showed  that  the 
cavity  of  the  woodpecker's  nest  was  being  gradually  filled  with  the 
"sawdust"  of  their  workings,  and  conspicuous  on  the  "sawdust" 
were  a  number  of  Syrphid  flies,  apparently  dead.  At  the  final 
investigation  at  the  beginning  of  October  about  a  hundred  and 
twenty  of  these  derelict  flies  were  found  in  the  central  cavity,  and 
as  there  were  certainly  not  more  than  six  wasps  at  work  at  any 
time,  and  as  two  were  early  captured  and  retained  for  identification, 
it  is  probably  safe  to  estimate  the  average  numbers  of  the  wasps 
responsible  for  discarding  them  at  five.  This  allows  twenty-four 
discarded  flies  to  each  wasp — about  six  hours  strenuous  labour  by 
each  insect  entirely  wasted!  As  wasp  No.  1  was  never  seen  to 
discard  a  captured  fly  this  phenomenon  was  apparently  attributable 
to  the  conditions  prevailing  inside.  There  all  the  burrows  com- 
menced with  a  horizontal  boring  at  the  junction  of  the  two  sections 
of  the  log,  at  some  little  distance  from  the  main  opening.  After 
alighting  at  the  main  entrance  they  had,  therefore,  either  to  fly 
across  or  to  crawl  round  the  central  cavity,  and  it  seems  as  though 
a  number  of  flies  had  been  accidentally  dropped.  It  would  be 
quite  in  keeping  with  what  has  been  observed  in  the  case  of  allied 
insects  that  a  wasp  which  had  accidentally  dropped  a  fly  should 
make  no  attempt  to  retrieve  it,  but  should  simply  go  away  and 
catch  another.  These  discarded  flies  were  in  any  case  very  useful 
as  evidence  of  the  particular  prey  selected  by  Crabro  cephalotes. 

At  the  beginning  of  October  some  of  these  flies  had  been 
reduced  to  fragments  by  other  predaceous  creatures,  but  of  113 
recognisable  specimens  60  were  S.  halteatus. 

My  friend  Mr  N.  D,  F.  Pearce  very  kindly  undertook  to  identify 
the  remainder  for  me  and  he  finds  among  them  five  species  of 
Syrphus,  three  of  Platychirus,  two  of  Melanostoma,  and  one  of 


Crabro  cephalotes  299 

Rhingia,  Catabomba  and  Helophilus  respectively.  No  family  of 
flies  except  the  Syrphidae  was  represented.  The  complete  list  is  as 
follows  : 

Syrphus  balteatus  60 

S.  luniger  5 

S.  vitripennis  4 

S.  corollae  4 

S.  auricoUis  3 

S.  albistr ictus  1 

Platychirus  albwianus  9  9 

P.  scutatus  ?  2 

P.  peltatus  1 

Melanostoma  mellinum  7 

M.  scalar e  ?  2 

Rhingia  campestris  13 

Catabomba  pyrastri  1 

Helophilus  pendulus  1 

113 

Early  in  October  the  log  was  thoroughly  explored,  and  an 
attempt  was  made  to  follow  out  the  windings  of  the  galleries, 
but  the  extreme  friability  of  the  decaying  heart-wood  made  this 
very  difficult. 

The  first  thing  that  struck  one  was  the  absence  of  any  attempt 
to  seal  or  mask  the  tunnels  which  were  entirely  open  to  any 
chance  intruder.  Indeed  a  family  of  wood-lice  was  found  three 
inches  down  the  tunnel  of  wasp  No.  1.  There  was  nothing  to  prevent 
any  enemy  from  entering.  While  at  work  the  wasps  had  never 
manifested  any  interest  in  other  insects  in  the  neighbourhood  of 
their  burrows,  nor  did  they  finally  make  any  provision  for  keeping 
them  out.  While  watching  the  operations  of  wasp  No.  1  a  few 
insects  had  been  seen  to  enter  the  tunnel,  including  Phoridae,  one 
of  which  was  secured,  and  a  Muscid  fly  ( ?  Tachina)  and  an  Ichneu- 
monid  which  unfortunately  evaded  capture. 

The  main  tunnels  were  clear,  and  penetrated  the  wood  for 
several  inches,  with  abrupt  turnings  on  no  definite  plan.  From 
these  proceeded  side  galleries  in  which  were  found  "  sawdust,"  the 
debris  of  flies,  and  the  brown  cocoons  containing  the  fully-fed  wasp 
larvae.  Sections  of  the  log  showed  that  these  were  dotted  here 
and  there  throughout  the  soft  heart- wood  precisely  hke  the  raisins 
in  a  Christmas  pudding. 


k 


300  Mr  Aston,  Neon  Lamps  for  Stroboscopic  Work 


Neon  Lamps  for  Stroboscopic  Work.  By  F.  W,  Aston,  M.A., 
Trinity  College  (D.Sc,  Birmingham),  Clerk-Maxwell  Student  of 
the  University  of  Cambridge. 

[Read  19  May  1919.] 

For  the  accurate  graduation  and  testing  of  revolution  indicators 
and  similar  technical  purposes  the  stroboscopic  method  is  probably 
the  most  reliable.  This  depends  on  the  fact  that  if  a  rotating  disc  is 
illuminated  N  times  per  second  by  very  short  flashes,  a  regular 
figure  drawn  symmetrically  on  the  disc  will  appear  at  rest  when 
the  number  of  revolutions  of  the  disc  per  second  is  some  exact 
multiple  or  submultiple  of  N  depending  on  the  number  of  sides  of 
the  regular  figure. 

The  value  of  N — in  practice  50 — can  be  s^t  and  easily  kept 
extremely  constant  by  the  use  of  an  electrically  driven  tuning-fork 
so  that  the  success  of  the  method  rests  principally  upon  the 
illuminating  flashes ;  its  accuracy  will  depend  upon  their  shortness 
of  duration  and  brightness;  its  convenience  as  a  practical  method 
upon  their  brightness  and  quality  as  affecting  the  eye  of  the 
observer. 

The  first  experiments  were  tried  with  naked  Ley  den  jar  sparks 
obtained  from  the  secondary  of  an  ordinary  ignition  coil,  the 
tuning-fork  being  introduced  into  the  primary  circuit  as  an 
interrupter.  These  showed  the  principle  of  the  method  to  be 
excellent  but  spark  illumination  left  much  to  be  desired;  it  was 
noisy,  feeble  in  intensity,  and  being  mostly  of  short  wave-length, 
caused  rapid  and  excessive  eye-strain  even  when  used  in  a  dark 
room. 

The  remarkable  properties  of  Neon  seemed  to  offer  an  almost 
ideal  solution  of  the  illumination  problem.  A  form  of  lamp  to 
replace  the  spark  was  therefore  devised  which  appeared  likely  to 
give  good  results  and  several  of  these  were  filled  from  the  author's 
stock  of  Neon  at  the  Cavendish  Laboratory.  The  success  of  these 
lamps  was  immediate,  eye-strain  disappearing  completely.  The 
present  paper  is  a  description  of  the  lamps  and  their  behaviour 
during  continuous  use. 

The  Form  of  Lamp. 

The  original  form  of  the  lamp,  which  it  has  not  been  found 
necessary  to  alter  materially,  is  shown  in  the  sketch.  As,  in  the 
discharge  in  Neon,  nearly  all  the  light  is  in  the  "Positive  Column" 
and  its  brightness  increases  with  the  current  density,  the  lamp 
was  designed  to  give  a  positive  column  as  long  and  narrow  as 


Mr  Aston,  Neon  Lamps  for  Stroboscopic  Work 


301 


possible  consistent  with  the  potential  available  in  the  spark,  and 
consists  essentially  of  two  relatively  large  spaces  containing  the 
electrodes  connected  by  a  very  long  capillary  tube  which  is  the 
counterpart  of  the  filament  in  an  ordinary  glow  lamp.  In  the  lamps 


flNOoe: 


-h 


C<^THOOE 

Neon  vacuum  lamp  for  Stroboscopic  work. 
Two-thirds  actual  size. 

in  use  the  filament  is  about  60  cm.  long  by  1  mm.  diameter  and  is 
coiled  up  inside  the  space  containing  the  anode.  This  was  done  for 
convenience  and  strength,  but  it  has  another  and  important 
advantage,  for  this  type  of  construction  is  strongly  unsymmetrical 
to  the  discharge,  allowing  it  to  pass  much  more  easily  in  the  direc- 


I 


302  Mr  Aston,  Neon  Lamps  for  Strohoscojnc  Work 

tion  indicated  in  the  figure  than  in  the  opposite,  hence  it  effectually 
stops  the  "reverse"  current  from  the  secondary  of  the  coil. 

Other  important  results  depending  on  the  length  of  the  fila- 
ment will  be  discussed  later,  it  should  be  roughly  one  hundred 
times  the  length  of  the  spark  the  coil  is  capable  of  giving  in  air 
when  running  on  the  tuning-fork  break. 

It  is  hardly  necessary  to  state  that  the  shape  into  which  the 
filament  is  wound  is  not  in  the  least  essential  and  could  be  varied 
to  any  extent  in  lamps  for  special  purposes. 

The  electrodes  are  of  aluminium  and  may  be  of  any  form  so 
long  as  they  are  not  too  small. 

Method  of  Filling  Lamps. 

As  Neon,  like  the  other  gases  of  the  Helium  group,  has  the 
remarkable  property  of  liberating  gas  from  aluminium  electrodes 
which  have  been  completely  run  in  for  other  gases,  the  operation 
of  filling  necessitates  the  contamination  of  a  comparatively  large 
volume  of  Neon,  so  that  this  can  only  be  done  economically  and 
conveniently  where  liquid  air  is  available  for  re-purifying. 

So  far  all  the  lamps  have  been  filled  on  the  author's  Neon 
fractionation  apparatus  at  the  Cavendish  Laboratory^,  The  gas 
for  filling  is  contained  in  charcoal  cooled  in  liquid  air.  A  quantity 
is  admitted  to  the  exhausted  lamp  which  is  then  sparked  at  a 
pressure  of  1  to  3  mm.  with  a  small  coil  for  a  time.  The  dirty  gas 
is  then  pumped  off  with  a  Toepler  mercury  pump,  a  fresh  supply 
of  pure  gas  admitted  and  the  tube  run  again.  These  operations 
are  repeated  until  spectroscopic  and  other  observations  show  the 
desired  conditions  of  purity  have  been  reached  and  are  not  altered 
seriously  by  prolonged  running.  The  full  charge  of  5  to  10  mm.  of 
gas  is  now  let  in  and  the  lamp  sealed  off.  The  whole  operation  takes 
about  3  hours,  three  lamps  being  filled  at  once.  The  pressure, 
purity  and  time  of  running  in  are  all  matters  of  some  nicety  as 
will  be  seen  from  consideration  of  the  life  of  the  lamp. 

Life  of  the  Lamps. 

Apart  from  accident  the  lamps  are  serviceable  until  the  pressure 
of  gas  within  them  becomes  too  low  for  the  spark  to  light  them 
adequately.  Their  life  appears  to  consist  of  two  distinct  periods, 
the  first  during  which  chemically  active  impurities  derived  from 
the  electrodes  and  walls  of  the  tube  are  being  slowly  and  completely 
eliminated  (at  least  as  far  as  a  spectroscopic  observation  goes)  and 
the  second  during  which  sputtering  of  the  cathode  takes  place  and 
the  inactive  Neon  itself  slowly  disappears  until  the  pressure  gets 
too  low  for  use.    During  the  first  period  the  luminosity  steadily 

1  V.  Lindemann  and  Aston,  Phil.  Mag.  sxxvii,  May  1919,  p.  527. 


Mr  Aston,  Neon  Lamps  for  Stroboscopic  Work  303 

improves,  remaining  almost  constant  afterwards  till  near  the  end 
of  the  second  period  when  it  rapidly  decreases. 

The  first  set  of  lamps  were  filled  with  very  carefully  purified 
Neon  at  1-2  mm.  pressure  and  run  till  sputtering  had  commenced 
before  being  used;  they  may  therefore  be  considered  to  have  had 
no  first  period  at  all.    These  lamps  had  a  life  of  500-1000  hours. 

Experiments  soon  showed  that  the  less  preliminary  running 
and  the  higher  the  pressure  of  filling  the  longer  the  life  would  be, 
but  on  the  other  hand,  if  the  preliminary  running  is  not  sufficient 
the  impurities  derived  from  the  electrodes  turn  the  light'  of  the 
lamp  a  dull  grey  and  render  it  absolutely  useless  and  pressures 
above  10  mm.  are  not  advisable  as  these  increase  the  spark 
potential  of  the  lamp  too  much. 

One  lamp  was  actually  so  nicely  balanced  in  these  respects 
that  though  it  became  grey  and  useless  after  about  1  hour's  use  it 
completely  recovered  its  original  brightness  after  a  day's  rest.  This 
is  clearly  a  case  of  carbon  compounds  being  given  off  by  the  elec- 
trodes while  running,  which  are  reabsorbed  on  standing  and  there 
is  little  doubt  that  were  it  worth  while  very  prolonged  running 
would  render  this  lamp  quite  satisfactory.  Very  slow  production 
of  gases  from  the  electrodes  is  advantageous,  as  prolonging  the 
first  period  of  the  life,  so  that  these  should  be  of  a  fairly  solid 
pattern. 

So  far,  the  best  results  have  been  obtained  from  a  batch  of 
lamps  filled  at  about  10  mm.  pressure,  some  with  pure  Neon, 
some  with  a  mixture  of  Neon  and  about  10  per  cent.  Helium. 

One  of  the  latter  had  a  working  life  of  well  over  3000  working 
hours.  Helium  disappearing  from  its  spectrum  after  the  first  few 
hundred. 

As  there  is  every  reason  to  assume  that  for  any  given  lamp  the 
life  is  determined  by  the  total  number  of  coulombs  passed  through 
it,  the  light  obtained  per  coulomb  should  be  arranged  to  be  a 
maximum.  This  will  be  the  case  when  the  filament  is  made  as  long 
as  possible,  consistent  with  the  potential  available  from  the  coil. 

Cause  of  Disappearance  of  Gas  from  the  Lamps. 
The  exhaustion  of  gas  by  continuous  running  has  long  been 
observed  in  the  case  of  spectrum  discharge  tubes.  It  is  doubtless 
allied  to  the  phenomenon  of  "Hardening"  in  X-ray  bulbs,  but 
difi'ers  from  the  latter  in  that  under  the  relatively  high  pressures 
in  spectrum  tubes,  and  the  Neon  lamps  under  consideration,  the 
mean  free-path  of  a  charged  molecule  is  so  small  that  it  can  only 
fall  freely  through  a  potential  of  a  few  hundred  volts  and  so  never 
attain  the  very  high  velocities  reached  in  the  X-ray  bulbs  which 
are  supposed  to  cause  the  gas  molecules  to  become  permanently 
embedded  in  the  glass  walls. 

VOL.  XIX.   PART  VI.  21 


304  Mr  Aston,  Neon  Lamfs  for  Strohoscopic  Work 

The  disappearance  of  gases  of  the  HeHum  group  in  spectrum 
tubes  is  invariably  associated  with  sputtering  of  the  electrodes 
which,  at  high  pressures,  only  takes  place  when  the  gas  is  spectro- 
scopically  free  from  chemically  active  gases.  It  is  generally  sup- 
posed that  the  gas  so  disappearing  remains  embedded  or  adsorbed 
in  the  layer  of  sputtered  aluminium  on  the  sides  of  the  tube  near 
the  cathode,  the  idea  of  true  chemical  combination  not  being 
acceptable  without  very  rigorous  proof. 

In  order  to  obtain  information  on  this  point,  a  completely  run 
out  specimen  of  the  first  batch  of  lamps,  which  was  of  course  very 
heavily  sputtered,  was  taken  for  test.  First  the  sputtered  cathode 
end  was  gradually  heated  to  near  the  softening  point  of  the  glass 
(when  it  cracked)  without  any  substantial  or  apparent  increase  in 
the  internal  pressure  of  Neon.  The  end  was  then  cut  ofi,  broken  into 
small  pieces  and  heated  in  a  quartz  tube  in  a  high  vacuum  apparatus 
provided  with  a  spectrum  tube.  At  a  temperature  about  the 
softening  point  of  the  glass  a  good  deal  of  gas  was  released  which 
showed  the  hydrocarbon  spectrum  (but  may  nevertheless  have 
contained  some  Neon  as  this  is  easily  masked) ;  this  gas  was  pumped 
off  and  on  heating  further  to  a  red  heat,  as  the  glass  started  to 
melt,  Neon  was  given  off,  the  spectrum  showing  quite  clearly. 

Apparatus  for  measurement  and  analysis  of  the  gas  so  released 
was  not  available,  but  it  is  hoped  to  repeat  this  interesting  experi- 
ment, which  shows  definitely  that  the  Neon  is  contained  either  in 
the  sputtered  aluminium  or  very  near  the  surface  of  the  glass  so 
that  it  is  released  by  heat. 

Use  of  other  Gases  instead  of  Neon. 

Ordinary  chemically  active  gases  give  very  feeble  illumination, 
CO  being  about  the  best.  Helium  gives  a  bright  discharge  but  not 
nearly  so  valuable  in  quality  for  visual  work  as  Neon ;  its  presence 
as  an  impurity  in  the  latter  gas  renders  the  discharge  more  rosy 
red  but  up  to  10  per  cent,  does  not  affect  its  brightness  seriously. 
Mercury  vapour  as  used  by  C.  T.  R.  Wilson  in  his  photography  of 
ionisation  tracks  would  probably  give  very  bright  flashes,  but  the 
fact  that  the  lamp  has  to  be  kept  very  hot  is  a  serious  objection. 

Reason  for  Superiority  of  Neon. 

The  brilliant  orange-red  glow  of  the  discharge  in  Neon  is  com- 
posed almost  entirely  of  lines  in  the  region  5700-6700  a.u.  and  is  in 
such  striking  contrast  to  sunlight  that  strohoscopic  observations 
can  even  be  done  in  broad  daylight  if  necessary,  the  ordinary 
appearance  of  the  rotating  disc  having  merely  a  grey  background 
added,  looking  bluish  by  contrast. 

The  actual  amount  of  light  radiated  per  unit  of  energy,  i.e. 


Mr  Aston,  Neon  Lamias  for  Stroboscopic  Work  305 

the  real  efficiency  of  the  discharge  in  Neon,  is  not  markedly  greater 
than  that  in  e.g.  mercury  vapour,  but  the  apparent  efficiency  is 
enormously  enhanced  by  the  fact  that  it  consists  so  largely  of  red 
light.  Victor  Henri  and  J.  L.desBancels  have  shown  ("  Photochemie 
de  la  Retine,"  Jl.  Phys.  Path,  xiii,  1911)  that  the  Fovea  Centralis  of 
the  eye  is  immensely  more  sensitive  to  red  light  than  the  outlying 
portions  of  the  retina^,  thus  a  Neon  lamp  as  a  source  of  general 
illumination  is  very  disappointing,  but  when  viewed  directly 
appears  surprisingly  bright.  As  the  spinning  disc  of  the  stroboscope 
subtends  a  comparatively  small  angle  the  Fovea  is  the  only  part 
of  the  observer's  eye  used  in  testing,  which  is  probably  the  reason 
for  the  eye  strain  with  the  spark. 

Nature  and  Duration  of  the  "  Working  Flash.'" 

If  one  analyses  the  flash  of  a  short  spectrum  type  Neon  tube  in 
a  rotating  mirror  it  is  seen  to  consist  of  two  separate  parts,  an 
extremely  short  flash  followed  by  a  flame  or  "arc."  The  first  is 
probably  due  to  the  simultaneous  ionisation  of  the  gas  throughout 
the  whole  length  of  the  tube,  the  second  to  the  further  carriage  of 
current  by  the  ions  formed  during  the  first.  The  structure  of  the 
latter,  which  appears  to  consist  of  bright  striations  travelling  from 
anode  to  cathode  at  velocities  of  the  order  of  that  of  sound  in  the 
gas,  is  of  great  theoretical  interest  and  is  at  present  under  investi- 
gation. Discussion  of  its  nature  is  needless  in  the  present  paper 
for  its  duration  being  of  the  order  of  thousandths  of  a  second  it  is 
useless  for  stroboscopic  work  and,  by  the  employment  of  a  suffici- 
ently long  filament  tube,  it  can  be  eliminated  altogether.  In  a 
lamp  properly  proportioned  to  the  power  of  the  coil  in  use  the 
v.'hole  energy  of  the  discharge  is  absorbed  in  the  first  flash.  In 
order  to  get  some  idea  of  the  duration  of  this  "working  flash"  the 
following  experiment  was  performed. 

A  plain  mirror,  silvered  outside  to  avoid  double  images,  was 
mounted  vertically  on  the  axis  of  a  large  centrifuge  and  the  image 
in  this  of  the  Neon  lamp  at  a  distance  of  3  metres  was  observed 
by  means  of  a  telescope  with  a  micrometer  eye-piece.  Each 
division  in  the  micrometer  subtended  4-2  x  10^^  radians  and  when 
the  centrifuge  was  running  at  3500  revolutions  per  minute  corre- 
sponded to  5-75  X  10~'  seconds. 

The  lamps  were  run  with  the  tuning-fork  attachment  used  in 
actual  testing  and  were  viewed  directly  and  also  through  ground 
glass  with  a  V-shaped  slit  to  be  certain  of  getting  the  effect  of  the 

'  The  difference  of  retinal  effect  between  red  and  green  light  can  be  easily  ob- 
served by  looking  at  an  ordinary  luminous  wrist  watch  in  the  faint  red  light  of 
a  photographic  dark  room.  On  shaking  the  watch  so  sluggish  is  the  green  light  in 
recording  its  position  on  the  retina  compared  with  the  red  that  the  figures  seem 
to  be  shaken  completely  off  the  dial,  giving  a  most  curious  and  striking  effect. 

21  —  2 


306  Mr  Aston,  Neon  Lamj)s  for  Sirohoscopic  Work 

total  duration  of  the  flash.  In  neither  case  was  the  fuzziness  of  the 
image  of  a  measurable  order.  After  careful  observation  under 
good  conditions  the  conclusion  of  three  observers  was,  that  it  was 
probably  less  than  one-tenth  of  a  division  and  certainly  less  than 
one-fifth.  This  gives  the  maximum  duration  of  the  working  flash 
as  one-ten-millionth  of  a  second,  so  that  it  can  be  taken  as  perfectly 
instantaneous  for  the  purpose  employed. 

Other  Technical  Afflications . 

Of  the  many  uses  besides  measuring  velocity  of  rotation  to 
which  Neon  lamps  may  be  put  with  advantage  in  engineering  and 
other  problems  it  is  sufficient  to  mention  two  in  which  they  have 
been  very  successful.  Any  rapidly  rotating  mechanism  such  as  an 
airscrew,  if  illuminated  by  a  lamp  the  break  of  which  is  operated 
mechanically  at  each  revolution,  will  appear  at  rest,  flicker  being 
small  at  speeds  well  over  1000  r.p.m.,  so  that  strains  or  movement 
of  parts  can  be  examined  with  great  accuracy  under  actual  working 
conditions. 

A  still  more  striking  effect  can  be  obtained  by  illuminating  a 
high  speed  internal  combustion  engine  by  a  lamp  whose  break  is 
operated  mechanically  at  e.g.  99  breaks  per  100  revolutions  of  the 
engine  shaft  by  the  use  of  a  creeping  gear.  The  engine  then  appears 
to  be  rotating  quite  smoothly  at  one-hundredth  its  normal  speed 
so  that  such  instructive  details  as  the  movements  of  the  valves 
and  springs,  the  bouncing  of  the  former  on  their  seats,  etc.,  can  be 
studied  with  ease. 

It  is  of  course  necessary  for  the  speed  of  rotation  to  be  fairly 
rapid  to  give  appearance  of  continuity  to  the  eye  and  in  conse- 
quence one  cannot  apply  this  method  to  the  analysis  of  such  a 
thing  as  the  movement  of  a  chronometer  escapement. 

As  the  technical  importance  of  Neon  lamps  is  rapidly  on  the 
increase  it  is  very  desirable  that  liquid  air  engineers  in  this  country 
should  consider  the  erection  of  a  fractionating  plant  for  recovering 
the  gas  from  the  air  (which  contains  -00123  per  cent,  by  volume) 
such  as  has  been  used  with  such  success  by  Mons.  Georges  Claude 
of  Paris,  to  whom  the  author  is  indebted  for  the  Neon  with  which 
these  experiments  were  performed. 


Mr  Harrison,  The  pressure  in  a  viscous  liquid  etc.         307 


The  ^pressure  in  a  viscous  liquid  moving  through  a  channel  unth 
diverging  boundaries.  By  W.  J.  Harrison,  M.A.,  Fellow  of  Clare 
College,  Cambridge. 

[Read  24  November  1919.] 

If  non- viscous  liquid  is  flowing  along  a  tube  having  a  cross- 
section  which  is  increasing  in  area  in  the  direction  of  flow,  the 
pressure  will  also  increase,  in  general,  in  the  same  direction.  On 
the  basis  of  this  remark  an  explanation  has  been  given  of  the 
secretory  action  of  the  kidneys.  The  author's  attention  was  drawn 
to  this  explanation  by  Dr  Ffrangcon  Roberts.  The  physiological 
aspect  of  the  question  and  a  more  detailed  numerical  consideration 
will  be  dealt  with  by  Dr  Roberts  and  the  author  in  a  separate 
paper. 

In  the  present  paper  two  problems  are  considered,  viz.  the  flow 
of  liquid  in  two  and  three  dimensions  when  the  stream  lines  are 
straight  lines  diverging  from  a  point. 

Two-dhnensional  'problem. 

Let  the  boundaries  of  the  channel  be  ^  =  ±  a,  where  (r,  d)  are 
two-dimensional  polar  coordinates.  The  motion  in  which  the  stream 
lines  are  straight  lines  passing  through  the  origin  has  been  ob- 
tained by  G.  B.  Jeffery^.  With  a  slight  change  of  notation  the 
results  of  his  solution  are  as  follows. 

Let  the  velocity  at  any  point  be  ujr,  where  m  is  a  function  of  d 
only.  Then 

2  A  ^^**   , 

U^  =  —  4:VU  —  V  -77i5-  +  a, 

dO^ 

where  v  is  the  kinematic  coefiicient  of  viscosity,  and  a  is  a  constant 
of  integration.   Whence 

u  =  —  2i^  (l  —  m^  —  m^k^)  —  Qvkhn^  sn^  {md,  k), 

where  k  and  m  are  constants,  which  may  be  determined  from  the 
conditions  that  u  must  vanish  at  ^  ==  ±  a,  and  that  the  total  rate 
of  flux  may  have  a  given  value.  Instead  of  the  latter  condition  it 
is  simpler  to  assume  that  the  velocity  is  given  for  ^  =  0,  i.e.  u  =  Uq 
for  6  =  0. 

Thus  the  conditions  are 

—  2v  {1  —  m^  —  m^k^)  =  Uq, 

(1  —  ni^  —  m^k^)  +  3kh)i^  sn^  {ma,  k)  =  0. 

1  Phil.  Hag.  (6),  vol.  xxix,  p.  459. 


308  Mr  Harrison,  The  pressure  in  a  viscous  liquid 

These  may  be  written 

m2  =  (1  +  uJ2v)l{l  +  B), 
'I  +  uJ2v\^     J  1  +  F 


sn' 


(^-^1?)-^ 


3F  (1  +  2i//wo) ' 

If  the  values  of  Uq  and  a  be  given,  the  last  equation  serves  for 
the  determination  of  h.  Writing  h-y  =  l/k,  the  equation  has  the 
same  form  in  k-^  as  in  Jc.  Hence,  if  k  is  a  solution,  l/k  is  also  a 
solution.  Therefore,  of  real  values  of  k,  it  is  only  necessary  to 
consider  such  that  satisfy  0  ^  ^  ^  1 . 

Treat  a  as  small,  and  assume  that  (  — = — %-^ —  )  a  is  also  small. 

V    l  +  k^    J 

(1  +  7(^2)2 

We  have  a^  =  ^-.^  .^    \,   , — tk— .  • 

3P  (2  +  2v/uq  +  uJ2v) 

The  least  value  of  a  for  a  given  value  of  Uq/2i',  if  k  is  real,  is  given 
hj  k  =  1.  In  this  case,  if  Uq/2p  =  1,  a^  =  ^,  a  =  -58.  This  value  of 
a  is  not  small  enough  for  the  approximation  to  hold  good.  Put 
^  =  1  and  2v/uq  =  1  in  the  original  equation,  and  we  find  a  =  "65, 
approximately.  For  smaller  values  of  a,  k  will  be  a  complex 
imaginary  quantity.  As  uJ2v  is  either  increased  or  decreased,  a 
real  value  for  k  can  be  obtained  for  smaller  values  of  a. 

It  will  be  found  sufiicient  for  the  purposes  of  the  present  paper 
to  restrict  the  consideration  of  the  solution  to  the  ranges  of  values 
of  a  and  Uq/2v  for  which  k  has  a  real  value.  We  proceed  to  discuss 
the  pressure  variation  in  the  case  for  which  k  is  real ;  the  variation 
in  the  case  for  which  k  is  complex  can  be  inferred  by  considerations 
of  continuity. 

Let  J)  be  the  mean  pressure  at  the  point  (r,  9)  in  the  liquid,  and 
p  its  density.  We  obtain  from,  the  two-dimensional  polar  equations 
of  motion 

u^  _       Idp       V  d'^u 
~^^  ~pdr^r^W' 
_       \  dy      2vdu 

^'^^^  ^-~2^-2^aP  +  ^(^) 

substituting  for  ^^  from  the  differential  equation  satisfied  by  u. 


Also  ^=J^+/(y) 


p       "'^ 


moving  through  a  channel  with  diverging  boundaries        309 

Hence  -  =  — - —  --^  +  C,  where  C  is  a  constant.    Now  the  lateral 
p        r^       zr^ 

stress  in  the  liquid  is  pee ,  where 

p  p  r 

Hence  pgg  is  independent  of  d,  and  is  the  normal  stress  (of  the 
nature  of  a  tension)  exerted  by  the  liquid  on  the  boundary.  If 
a  is  negative  the  normal  pressure  on  the  boundary  decreases  as 
the  channel  widens,  and  if  a  is  positive  the  normal  pressure 
increases. 

Now  by  substitution  of  the  solution  for  u  given  above  in  the 
differential  equation  satisfied  by  ii,  we  find 
a  =  4i;2  [_  1  +  m^  (1  -  F  +  J^)] 
=  iv^  [-!  +  (!  +  uJ2vf  (1  +  k^)/{l  +  k^fl 

(!)  Writing  a  =  0,  we  can  immediately  discriminate  between 
those  cases  for  which  the  pressure  on  the  boundary  decreases  and 
those  for  which  it  increases. 

If  a  =  0,  we  have 

1  +  uJ2p  -  (1  +  k^r/{i  +  k^)^, 

and  1  +  2i//mo  =  (1  +  k^f/{{l  +  k'^f  -  (1  +  k^)^ 

Hence        s„^  \(  A  +  ^Y.,  4  =  (1  +  P)*  -  (1 +^e)t 
iV(l  +  k^y/        )  sk^  (1  +  F)* 

The  following  diagram  shows  how  the  value  of  uJ2v  for  which 
Pffg  is  independent  of  r  varies  with  a,  for  those  cases  in  which  k  is 


310 


Mr  Harrison,  The  pressure  in.  a  viscous  liquid 


real.  It  clearly  indicates  that  when  a  is  small  the  critical  value  of 
Uq/2v  may  be  somewhat  large. 

If  a  >  7r/4,  the  lateral  pressure  increases  for  all  values  of  Uq. 

(2)  It  is  a  simple  matter  to  discuss  the  variation  of  the  pressure 
when  Uq/2i>  is  large.   We  have,  approximately 

uJ2v 


sn' 


1  +  k^ 
3F    = 


m"  = 


a  = 


1  +  P' 


(1  +  F)3    ' 

k  will  be  real  provided  F  >  |,   and,   corresponding  to  real 
values  of  k,  a  will  be  small. 

In  the  absence  of  viscosity,  so  that  u  =  Uq  for  all  values  of  d, 


1^69  = 


2r2 


C. 


1  +  k^ 
Thus  the  lateral  pressure  increases  at  a  rate  which  is  ■         ,^3 

of  the  rate  for  a  non- viscous  liquid. 

The  following  table  will  indicate  the  character  of  the  results 
when  k  is  real. 


uJ2. 

a 

{l+k^)/{l+k^r 

100 

10°  30' 
9°  30' 

•30 

•27 

1000 

1°    3' 
0°57' 

•30 

•27 

10,000 

0°    6' 

•28 

m 


For  larger  values  of  a  than  those  given  above,  and  for  the  corre- 
sponding values  of  uJ2v,  k  is  unreal. 

When  a  is  small  there  is  apparently  an  approximation  which 
JefEery  gives,  viz. 

u=  —  2v  {1  —  m^  —  m^k^)  —  Qvk^m'^d^, 


1 


moving  through  a  channel  with  diverging  boundaries        311 


where 

0  =  -  (1  -  m2  -  m^F)  - 

-  Skhn^a^, 

leading  to 

Qvm^  (1  —  m2),„„ 
=  «„  (1  -  ff'ja?). 

-«2) 

This  gives 

a  =  —  2vuja'^, 

and 

Pee             ^Uq    ,    ^, 

_        —             ..9...9.  +    ^  • 

/3  a^r" 

Thus  the  lateral  pressure  apparently  decreases  for  all  values  of 
Uq.   But  if 

Uq=  —  2v  {I  —  m^  —  m^k^), 

and  0  =  -  (1  -  w2  -  yn^)  -  Skhn^a^, 

we  have  Uq/2u  =  3khn^a^, 

and  therefore  ma  is  not  necessarily  small.  Hence  the  approxima- 
tion is  only  valid  for  values  of  Uq/2v  below  some  limiting  value. 
If  this  condition  be  satisfied  the  expression  for  pgg  given  above  is 
an  approximation  to  its  value  for  small  values  of  a. 

Three-dimensional  problem. 

Let  the  boundary  of  the  channel  be  ^  =  a,  where  {r,  9,  (f))  are 
polar  coordinates.  This  problem  has  been  considered  by  Prof. 
A.  H.  Gibson^.  In  his  solution  Cartesian  and  Polar  Coordinates 
are  confused,  and  he  assumes  that  the  stream  lines  are  straight  lines 
diverging  from  the  origin,  a  state  of  motion  which  is  impossible  if 
the  inertia  terms  are  retained  in  the  equations  of  motion,  as  he 
retains  them.  One  result  of  these  errors  is  that  in  his  solution  p  is 
a  function  of  6  although  the  preliminary  assumption  is  virtually 
made  that  p  is  independent  of  6.  His  expression  for  the  pressure 
appears  to  be  quite  wrong. 

Assume,  in  the  first  place,  that  the  stream  lines  are  straight 
lines  diverging  from  the  origin,  so  that  u  =  f{d)jr^,  v  =  0,  w  =  0. 
The  polar  equations  of  motion  reduce  to 


du  1  dp 

or  p  or 


dht      2  du     cot  6  du      1  c^u     2u 
dr^      r  dr         r^    dd      r^  dd^       r^ 


^  _       1  dp      2v  du 
pr^^^dd' 

0  =  -  -^  ^ 

p  d(f) 

1  Phil.  Mag.  (6),  vol.  xviii,  p.  36,  1909. 


312         Mr  Harrison,  The  pressure  in  a  viscous  liquid  etc. 

We  have 

i|  =  |-V^  [cot «./'+/"], 
1  dp  _2v  ., 

Hence  eliminating  f, 

^  +  ,-4  [/'"  +/"  cot  d  -/'  cosec2  e  +  6/']  =  0. 

Therefore  ff  =  0,  and 

/'"  +/"  cot  9-f'  cosec2  d+6f'  =  0. 
Hence/'  (6)  =  0,  and  the  boundary  conditions  cannot  be  satisfied^ 
since  u  becomes  independent  of  6. 

For  slow  motion,  or  any  motion  in  which  the  inertia  terms  can 
be  neglected,  we  have 

/'"  +/"  cot  e  -f  cosec2  d  +  6/'  =  0 (1). 

A  first  integral  is 

/"+/'cot0+6/+C  =  O  (2). 

The  solution  of  (2)  suitable  for  the  present  purpose  is 

/(6')  =  Z)(2-3sin2^)-iC. 
Let  f{e)-^u„     6=0, 

f{e)  =  0,       d^a. 
We  have  D  =  uJ3  sin^  a, 

(7  =  2  (2  —  3  sin^  a)  ^o/sin^  a. 
Hence  u  =  Uq  (sin^  a  —  sin^  6)/r^  sin^  a. 

Integrating  the  equations  of  motion,  we  have 

f=-|^(/'cot^+r)  +  i^x(^) 

^"d  P^=p+F,ir). 

Hence  ,^  ^ '^V(^)  +  ^^^  +  5, 

and  Vm^_19^_b. 

p  3  r* 

The  lateral  pressure  will  continually  increase  as  the  channel 

widens  if  C  be  negative,  that  is,  if  sin  a  >  (f)^,  or  a  >  54°  45'.  If 
a  <  54°  45',  for  sufficiently  small  values  of  Uq  the  pressure  will 
continually  diminish. 


Mr  Gray,  The  Effect  of  Ions  on  Ciliary  Motion  313 


The  Effect  of  Ions  on  Ciliary  Motion.  By  J.  Gray,  M.A., 
Fellow  of  King's  College,  Cambridge. 

[Read  10  November  1919.] 

The  ciliary  mechanism  of  the  gills  of  Mytilus  edidis  has  been 
described  by  Orton^.  There  are  at  least  four  distinct  sets  of  cilia. 
.  whose  movements  form  a  complex  but  highly  coordinated  system 
by  which  food  particles  are  filtered  from  the  sea-water  and  passed 
up  to  the  mouth.  This  coordinated  system  is  entirely  free  from 
any  nervous  control  and  continues  for  many  days  in  detached 
portions  of  the  gill.  These  gill  fragments  therefore  form  an 
admirable  material  for  the  physiological  study  of  ciliary  motion. 

The  effect  of  the  hydrogen  ion  on  ciliary  action  is  very  easily 
studied.  Normal  sea- water  has  a  Ph  of  about  7-8;  when  the  con- 
centration of  hydrogen  ions  is  increased  to  about  6-5  rapid  cessation 
of  movement  occurs.  In  sea-water  of  Ph  6-7  the  rate  of  ciliary 
movement  is  checked  at  first,  but  within  f-l|  hours  complete 
recovery  takes  place.  If  gill  fragments  whose  cilia  have  been 
stopped  by  the  more  acid  solution  are  returned  to  normal  sea- 
water,  complete  recovery  takes  place  in  less  than  20  minutes 
although  the  cilia  may  have  been  motionless  for  several  hours. 
A  large  number  of  experiments  have  been  performed  from  which 
it  is  clear  that  if  the  concentration  of  hydrogen  ions  is  only  slightly 
greater  than  normal,  the  cells  can  react  to  the  environment  and 
recovery  take  place  in  the  acid  solution.  In  stronger  acid,  however, 
recovery  only  takes  place  on  removing  the  gills  to  a  more  alkaline 
solution.  In  still  stronger  acid  the  cells  become  opaque  and  are 
killed. 

Gills  which  are  exposed  to  an  abnormally  high  concentration 
of  hydroxyl  ions  behave  in  a  remarkable  manner.  In  such  solu- 
tions ciliary  action  is  either  not  affected  at  all  or  proceeds  at 
an  abnormally  rapid  rate,  but  the  individual  cells  of  the  ciliated 
epithelia  break  away  from  each  other  and  move  about  in  the 
solution  owing  to  the  movement  of  their  cilia.  Since  such  cells 
are  no  longer  in  their  normal  environment,  it  is  impossible  to 
determine  any  upper  limit  of  hydroxyl  ions  which  will  permit 
normal  ciliary  action  to  go  on. 

Since  the  hydrogen  ion  has  a  most  marked  effect  on  ciliary 
activity,. it  is  necessary  to  adjust  the  hydrogen  ion  concentration 
of  all  artificial  solutions  during  a  study  of  the  effects  of  various 
salts  on  ciliary  action.  In  the  case  of  the  salts  of  the  alkali  metals 
this  is  satisfactorily  performed  by  the  addition  of  an  appropriate 

1  Journ.  Marine  Biol.  Assoc,  vol.  ix,  p.  444  (1912). 


314  Mr  Gray,  The  Effect  of  Ions  on  Ciliary  Motion 

buffer  such  as  sodium  bicarbonate.  In  the  case  of  the  salts  of  the 
alkaline  earths  it  is  impossible  to  obtain  pure  isotonic  solution  of 
the  same  hydrogen  ion  concentration  as  sea-water,  and  it  is  there- 
fore necessary  to  compare  the  effects  of  the  pure  solutions  with 
that  of  sea-water  whose  hydrogen  ion  concentration  is  abnormally 
high. 

A  number  of  experiments  have  been  performed  which  prove 
that  sodium,  potassium,  calcium  and  magnesium  are  all  necessary 
to  maintain  gill  fragments  in  a  normal  state  of  ciliary  activity 
for  a  protracted  period,  viz.  four  days.  If  one  or  more  metals  are 
omitted,  the  individual  cells  of  the  ciliated  epithelia  show  the  same 
disruptive  phenomenon  as  in  sea-water  of  abnormally  high  con- 
centration of  hydroxyl  ions.  Solutions  containing  only  one  metal 
show  this  phenomenon  to  a  very  marked  degree  although  they 
may  be  more  acid  than  normal  sea-water;  the  effect  of  solutions 
containing  two  metals  is  less  marked  than  that  of  solutions  contain- 
ing only  one  metal,  but  more  marked  than  that  of  solutions  con- 
taining three  metals.  No  evidence  was  obtained  of  specific  ion 
action  or  of  antagonistic  action  between  monovalent  and  divalent 
ions. 

These  experiments  afford  another  example  of  the  intense  action 
of  the  hydrogen  ion  upon  physiological  activity  and  of  its  reversible 
nature  if  the  acid  treatment  is  not  too  severe.  The  same  action  of 
acids  is  found  in  the  activity  of  the  heart  and  in  the  movement  of 
spermatozoa. 


Mr  Saunders,  Photosynthesis  and  Hydrogen  Ion  Concentration   315 


A  Note  on  Photosynthesis  and  Hydrogen  Ion  Concentration.  By 
J.  T.  Saunders,  M.A.,  Christ's  College. 

[Read  10  November  1919.] 

Last  April  (1919)  I  was  testing  the  hydrogen  ion  concentration 
of  the  water  of  Upton  Broad,  a  small  broad  in  Norfolk.  I  had 
determined  the  hydrogen  ion  concentration  of  the  water  of  the 
broad  itself  to  be  8-3  and  I  found  this  varied  very  Httle  whether 
the  water  was  taken  from  the  surface  or  the  bottom,  from  near  the 
edge  or  the  centre  of  the  broad.  The  determination  of  the  hydrogen 
ion  concentration  was  made  by  the  use  of  standard  solutions  and 
indicators  as  recommended  by  Clark  and  Lubs. 

When  however  the  water  in  the  shallow  lodes  and  ditches 
surrounding  the  broad  was  tested,  great  variations  in  the  hydrogen 
ion  concentration  occurred.  The  water  became  more  acid  as  soon 
as  the  broad  was  left  and  the  ditches  entered.  At  one  end  of  the 
broad  where  the  water  was  shallow,  not  more  than  18  inches  deep, 
and  when  there  was  no  wind  to  mix  it  with  the  open  waters  of  the 
broad  which  was  6  feet  deep,  the  hydrogen  ion  concentration 
would  fall  to  8' 15.  In  the  lode  itself  the  hydrogen  ion  concentration 
was  7-65.  After  boiling  and  rapidly  cooling,  water  from  the  middle 
of  the  broad  and  from  the  shallows  both  showed  a  hydrogen  ion 
concentration  of  8'4,  while  that  from  the  lode  after  the  same  treat- 
ment was  8"15. 

At  one  point  in  the  lode,  however,  I  found  surprising  varia- 
tions. Dippings  of  water  from  the  same  place  gave  readings  of  the 
hydrogen  ion  concentration  varying  from  7-7  to  8-6.  At  this  point 
there  was  a  certain  amount  of  Spirogyra  growing  and  I  found  that 
if  I  took  water  from  the  centre  of  a  mass  of  Spirogyra  I  could  get 
a  reading  as  high  as  9-0. 

I  took  some  of  the  Spirogyra  back  with  me  and  placed  it  in 
test-tubes  in  tap-water  which  I  coloured  with  indicator  solutions. 
The  hydrogen  ion  concentration  was  7-2  at  the  commencement  of 
the  experiment.  After  standing  the  test-tube  in  a  window  in  sun- 
light the  hydrogen  ion  concentration  rose  after  an  hour  to  8-6  and 
in  two  hours  the  phenolphthalein  indicator  had  turned  bright 
pink,  indicating  a  hydrogen  ion  concentration  of  more  than  9-0. 
I  had  no  standard  solutions  with  me  which  I  could  use  to  test 
higher  values  than  9-0  so  that  I  was  unable  to  determine  accurately 
the  ultimate  result.  I  left  the  test-tubes  until  the  next  morning, 
when  I  found  the  hydrogen  ion  concentration  had  fallen  to  7-6. 
After  again  placing  the  test-tubes  in  sunlight  the  hydrogen  ion 
concentration  rose  above  9-0. 

On  my  return  to  Cambridge  I  repeated  these  rough  experi- 
ments. It  is  easy  to  prove  that  the  rise  in  alkalinity  is  not  due  to 
alkali  dissolved  out  of  the  glass,  nor  is  it  due  alone  to  the  abstrac- 


316  Mr  Saunders,  Photosynthesis  and  Hydrogen  Ion  Concentration 


tion  of  the  dissolved  carbon  dioxide  out  of  the  water.  The  hydrogen 
ion  concentration  of  the  Cambridge  tap-water  which  I  used  for 
these  experiments  was  7-15  when  the  water  was  tested  immediately 
after  being  drawn  from  the  tap.  On  standing  at  a  temperature  of 
13°  C.  the  hydrogen  ion  concentration  rises  to  l-i.  After  boihng 
and  rapidly  cooling  the  hydrogen  ion  concentration  was  7-9  and 
bubbling  through  air  free  from  carbon  dioxide  produced  the  same 
result.  By  incubating  tap-water  for  36  hours  at  a  temperature  of 
40°  C.  and  then  cooling  the  hydrogen  ion  concentration  could  be 
made  to  rise  to  8-15,  but  in  no  case  did  the  value  of  the  control 
tap-water  approach  near  that  of  the  tap-water  containing  Spiro- 
gyra  filaments. 

The  following  is  a  record  of  a  typical  experiment.  The  Spirogyra 
was  placed  in  25  c.c.  of  tap- water  in  a  boiling  tube  and  exposed  to 
light  at  a  window.  Control  boiling  tubes  containing  tap-water 
only  were  used.  All  these  tubes  were  half  immersed  in  a  glass  bowl 
of  running  water  so  that  the  temperature  was  maintained  fairly 
constant. 


Hydrogen  Ion 

concentration 

Date 

Time 
(G.M.T.) 

Temp. 

Remarks 

1 

Control 

Spirogyra 

1.  V.  19 

11-10  a.m. 

14-0°  C. 

7-15 

7-15 

DuU  day. 

12-10  p.m. 

13-0°  C. 

7-4 

8-3 

1.10  p.m. 

12-5°  C. 

7-4 

8-6 

2.10  p.m. 

12-5°  C. 

7-4 

8-6 

3.10  p.m. 

12-5°  C. 

7-4 

8-8 

5.30  p.m. 

13-0°  C. 

7-4 

8-5 

I  have  tried  using  Elodea  instead  of  Spirogyra  and  it  gives 
much  the  same  result. 

Both  in  darkness  and  in  daylight  the  contents  of  the  living  cell 
of  Spirogyra  show  an  acid  reaction  when  stained  with  neutral 
red.  When  Spirogyra  is  killed  by  heating  to  40°  C.  and  then  placed 
in  tap-water  the  hydrogen  ion  concentration  falls  considerably 
since  the  cell  membranes  are  broken  or  dead  and  the  contents  of 
the  cell  are  now  free  to  pass  out  into  the  water. 

In  a  large  pond  the  mass  of  the  plants  in  proportion  to  the 
water  is  not  sufficiently  great  to  affect  the  hydrogen  ion  concen- 
tration very  much.  I  have  however  found  slight  variations.  On 
one  occasion  I  noticed  a  fall  in  the  hydrogen  ion  concentration  of 
0-1  after  several  dull  days  and  a  subsequent  rise  of  0-2  after  sunny 
days.  This  variation  may  possibly  be  due  in  some  degree  to  the 
photosynthetic  activity  of  the  plants  present. 


Mr  Aston,  Distribution  of  intensity  317 


The  distribution  of  intensity  along  the  positive  ray  parabolas  of 
atoms  and  molecules  of  hydrogen  and  its  possible  explanation.  By 
F.  W.  Aston,  M.A.,  Trinity  College  (D.Sc,  Birmingham).  Clerk- 
Maxwell  Student  of  the  University  of  Cambridge. 

[Read  19  May  1919.] 

No  one  working  with  positive  rays  analysed  by  Sir  J.  J. 
Thomson's  method  can  fail  to  notice  the  very  remarkable  intensity 
variation  along  the  molecular  and  atomic  parabolas  described  by 
him  under  the  term  '  beading.'  It  will  be  sufficient  for  the  reader 
to  refer  to  Plate  III  of  his  monograph  on  the  subject  {Rays  of 
positive  electric,  p.  52)  to  realise  how  striking  these  can  be. 
Beadings  at  points  corresponding  to  energy  greater  than  the  normal 
have  been  quite  satisfactorily  accounted  for  by  multiple  charges 
{I.e.,  p.  46),  but  the  ones  with  which  this  paper  is  concerned  have 
a  smaller  energy  than  the  normal,  actually  half,  and  fractional 
charges  are  presumably  impossible.  Nevertheless  they  seem 
capable  of  a  simple  explanation  and  an  opportunity  of  putting 
this  to  the  test  occurred  recently  while  making  some  experiments 
to  determine  the  best  form  and  position  of  the  cathode  pre- 
liminary to  the  design  of  an  apparatus  to  carry  the  analysis  to 
higher  degrees  of  precision. 

The  observations  were  made  with  an  apparatus  essentially  of 
the  form  now  well  known  {I.e.,  p.  20)  the  discharge  tube  being 
arranged  to  be  removable  with  the  minimum  trouble  to  change 
or  move  the  cathode.  As  no  camera  suitable  for  photographic 
recording  was  immediately  available  or  necessary  a  willemite 
screen  and  visual  observation  was  employed.  This  form  has  many 
obvious  disadvantages  and  in  addition,  owing  to  the  enormous 
difEerence  in  sensitivity  between  the  parabolas  of  hydrogen  and 
those  due  to  heavier  elements  the  latter  can  only  be  seen  with 
difficulty.  It  has  however  one  notable  advantage,  namely  that 
sudden  and  even  momentary  changes  in  intensity  can  be  observed 
and  correlated  in  time  with  changes  in  the  discharge  or  in  the 
intensity  of  other  lines.  As  no  accurate  measurements  were 
intended  a  large  canal  ray  tube  was  employed  so  that  the  H^  and 
H^  parabolas  could  be  easily  seen  even  with  the  less  effective  types 
of  cathode. 

It  was  soon  realised  that  the  appearance  on  the  screen  was  in 
general  the  sum  of  two  superposed  effects  which  could  be  only 
unravelled  like  the  writings  on  a  palimpsest  by  eliminating  one  of 
them.  This  by  good  fortune  it  was  found  possible  to  do  under 
certain  conditions.    For  the  sake  of  clearness  it  is  proposed  to 


I 


318 


Mr  Aston,  Distribution  of  intensity 


consider  these  two  extreme  types  and  their  explanation  before 
going  on  to  describe  the  conditions  under  which  they  may  be 
attained  or  approached.  In  the  diagrams  the  fields  of  electric  and 
magnetic  forces  are  horizontal  and  such  that  positive  ions  will  be 
deflected  to  the  right  and  up,  negative  ones  to  the  left  and  down. 
Brightness  is  roughly  indicated  by  the  width  of  the  parabolic  patch 
drawn. 


O 


Fig.  1.     Atomic  Type. 


Atomic  type  of  discharge. 

Fig.  1  illustrates  the  first  or  'Atomic'  type  in  which  apparently 
the  whole  of  the  discharge  is  carried  up  to  the  face  of  the  cathode 
by  ions  of  atomic  mass.  Those  which  pass  through  the  fields 
without  collision  produce  the  true  primary  streak  on  parabola 
m=l,  the  head  of  which  corresponds  in  energy  to  that  obtained 
by  the  charge  e  falling  through  the  full  potential  of  the  discharge. 
Now  the  pressure  in  the  canal  ray  tube  is  never  negligible  being  on 
the  average  at  least  half  that  in  the  discharge  tube,  and  the 
ionisation  along  its  length  very  intense  so  that  in  passing  through 
it  a  large  number  will  collide  with  electrons,  atoms  or  molecules. 
The  collision  and  capture  of  a  single  negative  electron  will  result 


along  positive  ray  parabolas  of  atoms  and  molecules  of  hydrogen     319 

in  a  neutral  atom  striking  the  screen  at  the  central  undeflected 
spot  0  while  the  capture  of  two  will  cause  the  faint  negative 
parabolic  streak  a^  as  has  already  been  described  (I.e.,  p.  39). 

But  besides  these  forms  of  collision  by  which  the  velocity  of 
the  atom  is  practically  unaffected  there  is  distinct  evidence  that 
it  may  collide  with  and  capture  another  hydrogen  atom.  If  the 
atom  struck  is  negatively  charged  the  resulting  molecule  will 
strike  the  central  spot  but  if  it  is  neutral  and  the  collision  is 
inelastic  the  resulting  positive  ray  will  have  the  same  momentum 
(the^  atom  struck  being  relatively  at  rest)  but  double  the  mass  so 
that  it  will  strike  the  molecular  parabola  at  a  point  the  same  height 
above  the  JT-axis  as  would  the  atom  which  generated  it.  Molecular 
rays  formed  in  this  manner  will  therefore  form  the  streak  b^ 
which,  allowing  for  the  geometrical  difference  in  the  curves  will 
show  a  similar  distribution  of  intensity  to  a^.  Collision  with  a 
positively  charged  atom  wiU  obviously  be  unlikely  to  result  in 
capture  and  those  with  heavier  atoms  will  be  referred  to  later. 
It  is  to  be  noted  in  connection  with  the  brightness  of  these 
secondary  streaks  a^  and  63'  which  may  conveniently  be  called 
'satellites'  to  distinguish  them  from  the  'secondary  lines'  already 
fully  described  {I.e.,  p.  32),  that  a^  is  always  very  much  fainter  than 
its  primary  but  b^  can  be  equally  bright. 

This  atomic  type  of  discharge  with  its  pendant  bright  arc  on 
the  molecular  parabola  corresponding  to  similar  momentum  and 
half  normal  energy  is  most  beautifully  illustrated  in  Fig.  29  of 
Plate  III  already  referred  to.  It  was  this  photograph  which 
suggested  the  above  theory  of  its  explanation. 

Molecular  type  of  discharge. 

The  extreme  form  in  which  the  whole  discharge  is  carried  up 
to  the  cathode  by  ions  of  molecular  mass  is  unattainable  so  far 
in  practice  and  is  probably  impossible  but  its  share  in  the  illumina- 
tion of  the  screen  can  be  deduced  by  eliminating  the  superimposed 
atomic  type  and  is  indicated  in  Fig.  2, 

The  principal  feature  is  a  short  and  very  bright  spot  of  light  b^ 
on  the  molecular  parabola  at  the  point  corresponding  in  energy 
to  a  fall  through  the  full  potential  of  the  discharge.  It  will  be 
shown  that  all  the  ions  causing  this  are  probably  generated  in  the 
negative  glow.  Besides  this  there  are  two  symmetrical  and  equally 
bright  positive  and  negative  satellite  patches  ag  ^^^  (^2  on  the 
atomic  parabola  but  of  half  the  normal  energy.  The  proposed 
explanation  of  these  is  somewhat  similar  to  that  considered  by 
Sir  J.  J.  Thomson  {I.e.,  p.  94)  and  is  as  follows.  The  collision  with 
and  capture  of  a  single  negative  electron  by  a  positively  charged 
molecule  will  not  necessarily  merely  neutralise  it  and  cause  it  to 

VOL.  XIX.  PART  VI.  22 


320 


Mr  Aston,  Distribution  of  intensity 


hit  the  central  spot  0  but  may  result  in  it  splitting  into  two  atoms 
one  with  a  positive  one  with  a  negative  charge.  The  energy  of 
impact  may  be  itself  capable  of  causing  this,  if  not  some  other 
cause,  e.g.  radiation,  may  effect  the  dissociation.  In  any  case  it 
would  give  exactly  the  observed  result,  i.e.  two  bright  patches 
lying  symmetrically  on  the  extension  of  the  line  joining  the 
primary  spot  to  the  origin  at  twice  its  distance  from  the  latter, 
corresponding  to  half  the  mass  but  the  same  velocity. 


•^ 

__i 

^ 

/c. 

Fig.  2.     Molecular  Tyi^e. 

The  general   appearance   on  the  screen  when  both  types   of 
discharge  are  present  is  indicated  in  Fig.  3. 


Effect  of  different  forms  of  cathode. 

Experiments  were  performed  with  plane,  concave  and  convex 
cathodes.  Convex  cathodes  are  the  least  efficient  in  producing 
bright  effects  but  give  the  molecular  type  with  the  least  atomic 
blurring.  Concave  ones  are  most  efficient  and  throw  the  maximum 
energy  into  the  atomic  type  which  can  be  obtained  practically 
pure  with  them  under  a  moderate  range  of  conditions.   The  original 


along  positive  ray  parabolas  of  atoms  and  molecules  of  hydrogen    321 


shape  of  cathode  {I.e.,  p.  20)  may  be  said  in  a  sense  to  combine 
both  forms  and  was  designed  to  give  long  and  bright  parabolas 
at  the  same  time  allowing  the  discharge  to  pass  easily  at  very  low 
pressures.  The  present  results  however  lead  one  to  recommend  a 
concave  cathode  similar  to  those  used  in  X-ray  focus  tubes  but 
pushed  further  forward  into  the  neck  of  the  bulb,  for  though  this 
form  requires  a  rather  higher  pressure  this  objection  is  more  than 
counterbalanced  by  the  great  increase  in  efficiency.  Plane  cathodes, 
as  was  expected,  give  effects  midway  between  the  other  forms. 


O 


/ 


Fig.  3.     General  Type. 

Under  very  exact  conditions  of  pressure,  etc.  it  is  possible  to 
obtain  the  pure  atomic  type  with  plane  cathodes  but  no  conditions 
have  yet  been  found  under  which  convex  ones  will  give  it. 

These  results  seem  to  indicate  that  atomic  ions  are  formed  by 
the  passage  of  the  stream  of  cathode  rays  through  the  Crookes 
dark  space  molecular  ones  tending  rather  to  be  formed  in  the 
negative  glow.  The  axial  intensity  of  the  cathode  stream  is 
enormously  increased  by  the  concavity  of  the  cathode  while  that 
of  the  negative  glow  does  not  appear  to  be  affected  to  anything 
like  the  same  extent. 

22 2 


322  Mr  Aston,  Distribution  of  intensity 

Behaviour  during  change  of  pressure. 

The  pressure  in  a  freshly  set  up  bulb  always  increases  with 
running  owing  to  the  liberation  of  gas  by  heat  etc.  so  that  the 
changes  due  to  gradual  alteration  of  pressure  can  be  observed 
most  conveniently  by  exhausting  highly,  starting  the  coil  and 
watching  the  events  on  the  screen.  Thus  using  a  concave  cathode 
of  about  8  cms.  radius  of  curvature  set  just  in  the  neck  of  the 
discharge  bulb  the  following  sequence  of  events  was  observed. 
At  very  low  pressures  with  a  potential  of  about  50,000  volts  the 
parabolas  are  very  faint  but  correspond  to  the  general  type,  the 
primary  streak  a^  and  spot  h-^  being  much  brighter  than  their 
satellites  (doubtless  due  to  few  collisions).  As  the  pressure  rises 
the  discharge  becomes  curiously  unsteady  the  spots  on  the  screen 
become  much  fainter  and  change  with  flickering  into  the  pure 
atomic  type  (Fig,  1),  6i  having  practically  disappeared.  This  form 
of  discharge  which  is  evidently  abnormal  lasts  for  a  certain  time 
depending  on  the  rate  of  increase  of  pressure.  Then  with  absolute 
suddenness  h-^  flashes  out  intensely  bright  and  with  it  appear  at 
the  same  instant  its  satellites  a^  and  a^.  At  the  same  time  the 
current  through  the  bulb  increases,  the  discharge  settles  down  and 
the  negative  glow  makes  its  appearance.  As  far  as  it  was  possible 
to  judge  the  satellites  a^,  and  a^  are  of  equal  brightness  and  generally 
much  brighter  than  the  negative  atomic  satellite  «!. 

The  appearance  of  the  discharge  bulb  while  the  pure  atomic 
type  is  shown  on  the  screen  is  difficult  to  describe  but  quite 
characteristic  and  different  from  the  general.  Near  its  critical 
upper  limit  of  pressure  it  was  found  possible  to  effect  the  change 
to  the  general  type  by  bringing  a  magnet  near  the  cathode  and 
so  disturbing  the  discharge.  On  removing  the  magnet  the  discharge 
at  once  reverted  to  the  atomic  type.  This  form  of  controlled 
change  from  the  one  to  the  other  gave  an  excellent  opportunity 
of  testing  the  invariable  association  between  the  primary  spots 
and  their  appropriate  satellites. 

Possible  cause  of  disappearance  of  primary  molecular  rays. 

It  is  unlikely  that  change  of  pressure  is  itself  the  determining 
factor  in  the  disappearance  of  the  molecular  type.  This  seems  to 
be  due  to  some  disturbance  in  the  discharge  by  the  cathode  stream 
(not  caused  by  the  diffuse  one  given  by  a  convex  cathode)  which 
makes  the  formation  of  the  negative  glow  impossible. 

The  facts  so  far  may  be  brought  into  line  fairly  well  by  the 
somewhat  speculative  assumption  that  molecular  rays  can  only 
originate  freely  in  parts  of  the  discharge  where  the  electric  force 
is  very  small,  e.g.  the  negative  glow,  ionisation  by  more  violent 


alo'ng  positive  ray  parabolas  of  atoms  and  tnolecules  of  hydrogen    323 

means  in  strong  fields  tending  to  cause  simultaneous  disruption 
of  the  molecule  into  its  atomic  constituents.  This  agrees  with  the 
observed  fact  that  in  general  molecular  arcs,  or  at  least  true 
primary  molecular  arcs,  are  shorter  than  atomic  ones.  It  would 
also  mean  that  a  very  short  arc  infers  as  origin  a  molecule  capable 
of  disruption.  If  this  is  so  it  offers  interesting  confirmatory 
evidence,  if  such  were  needed,  that  the  substance  X^  is  molecular 
as  this  body  often  makes  its  appearance  on  the  photographic  plate 
as  a  short  arc. 

Effects  with  heavier  elements. 

The  inelastic  collision  of  a  hydrogen  atomic  positive  ray  with 
the  atom  of  a  heavy  element  would  clearly  result  in  the  formation 
of  a  molecular  ray  of  such  low  velocity  that  it  might  not  be 
detected  by  a  screen  or  plate  and  would  in  any  case  be  deflected 
completely  off  the  ordinary  photograph. 

The  visual  evidence  on  the  screen  although  faint  leaves  little 
doubt  that  the  formation  of  satellite  arcs  also  takes  place  by 
atoms  of  heavier  elements  colliding  to  form  molecules.  There  is 
also  some  evidence  of  this  in  many  of  the  photographs,  thus  in 
Fig.  26  (I.e.,  p.  46)  taken  with  oxygen  all  four  maxima  are  suggested. 
In  Fig.  17  (p.  26)  the  satellite  on  the  molecular  parabola  caused 
by  the  capture  of  oxygen  atoms  by  carbon  atomic  rays  (or  vice 
versa,  but  this  is  less  likely)  is  unmistakable,  in  fact  attention  is 
called  in  the  text  to  this  remarkable  increase  in  brightness. 

Should  the  above  theory  of  collision  with  capture  prove 
correct  the  formation  of  compound  molecules  by  this  means  opens 
an  extremely  interesting  field  of  chemical  research.  Another 
important  question  raised  is  in  what  form  the  energy  of  the 
collision  is  radiated  off  by  the  rapidly  rotating  doublet  formed. 

In  conclusion  the  author  wishes  to  express  his  indebtedness  to 
the  Government  Grant  Committee  for  defraying  the  cost  of  some 
of  the  apparatus  used  in  these  experiments. 


324  Sir  Joseph  Larmor 

Gravitation  and  Light.  By  Sir  Joseph  Larmor,  St  Johirs 
College,  Lucasian  Professor. 

[Read  26  January  1920.] 

1.  Newton's  provisional  thoughts  on  the  deep  questions  of 
physical  science  were  printed  at  the  end  of  the  second  edition 
of  the  Opticks  in  1717.  As  he  explains  in  the  Preface  "  .  .  .at  the 
end  of  the  Third  Book  I  have  added  some  questions.  And  to  shew 
that  I  do  not  take  Gravity  for  an  Essential  Property  of  Bodies, 
I  have  added  one  Question  concerning  its  Cause,  chusing  rather  to 
preface  it  by  way  of  a  Question,  because  I  am  not  yet  satisfied 
about  it  for  want  of  Experiments."  In  the  first  and  next  following 
Queries  he  gives  formal  expression  to  the  idea  that  "Bodies  Act 
upon  Light  at  a  distance  and  by  their  action  bend  its  Rays.  ..." 

What  was  thus  propounded  in  general  terms  as  an  explanation 
of  the  diffraction  of  light  in  passing  close  to  the  edge  of  an  obstacle, 
assumed  a  more  definite  but  different  form  in  the  hands  of  the 
physically-minded  John  Michell*;  in  Phil.  Trans.  1767  he  insisted 
that  the  Newtonian  corpuscles  of  light  must  be  subject  to  gravita- 
tion like  other  bodies,  therefore  that  the  velocities  of  the  corpuscles 
shot  out  from  one  of  the  more  massive  stars  vrould  be  sensibly 
diminished  by  the  backward  pull  of  its  gravitation,  and  thus  that 
they  would  be  deviated  more  than  usual  by  a  glass  prism,  a  supposi- 
tion which  he  proposed  to  test  by  experiment.  He  also  speculated 
that  the  scintillation  of  the  stars  might  be  due  to  the  small  number 
of  corpuscles  which  reach  the  eye  from  a  star,  amounting  perhaps 
to  only  a  few  per  second. 

The  forces,  of  molecular  range,  that  would  have  to  be  con- 
cerned, on  the  lines  of  Newton's  Query,  in  the  diffraction  of  light 
would  be  of  course  enormously  more  intense  than  gravitation :  but 
the  other  Newton-Michell  theory  of  the  gravitation  of  light  rays 
is  paralleled  in  both  its  aspects  with  curious  closeness  in  certain 
modern  physical  speculations. 

It  will  be  observed  that  this  notion  of  light  being  subject  to 
gravitation  makes  its  velocity  exceed  the  limiting  velocity  c,  which 
on  electrodynamic  theory  could  not  be  attained  by  any  material 
body.  But  there  need  not  be  a  discrepancy  there:  for  the  limit 
arises  because  a  material  body  is  supposed  to  acquire  more  and 
more  inertia,  belonging  to  energy  of  its  motion,  without  limit  as 
its  velocity  increases,  whereas  the  quantum  of  energy  in  the  hypo- 
thetical light-bundle  presumably  would  remain  sensibly  the  same — - 
at  any  rate  we  would  be  free  to  make  hypotheses  in  absence  of 
any  knowledge. 

*  See  Memoir  of  John  Michell  (of  Queens'  College),  by  Sir  A.  Geikie,  Cambridge 
Press,  1918. 


Gravitation  and  Light  325 

Forty  years  a.oo  there  was  a  phase  of  strong  remonstrance  in 
this  country  against  the  famihar  uncritical  use  of  the  phrase 
centrifugal  force.  The  implication  was  that  the  term  force  should 
be  restricted  to  intrinsic  unchanging  forces  of  nature,  which  are 
determined  physically  by  the  mutual  configuration  of  the  system 
of  bodies  between  which  they  act:  these  forces  are  then  held 
responsible  for  the  accelerative  effects  specified  by  the  Newtonian 
second  law  of  motion.  In  this  sense,  centrifugal  force  so-called 
would  not  be  a  force  of  nature,  but  would  be  the  reaction  postulated 
in  the  scheme  of  the  Newtonian  third  law  to  balance  an  imposed 
centripetal  acceleration. 

This  formative  principle,  the  Newtonian  third  law,  of  balance 
everywhere  between  appHed  forces  and  reactions  against  palpable 
changes  of  motion,  as  amplified  in  the  Scholium  an]iexed  to  it — 
which  so  widely  reached  forward  towards  modern  theory  as 
Thomson  and  Tait  especially  have  remarked — would  then  assert 
that  the  forces  of  nature  that  act  on  the  framework  of  a  material 
body  and  the  forces  of  reaction  that  are  thereby  induced  in  it, 
form  together  a  system  of  forces  that  preserve  statical  equilibrium 
in  relation  to  the  constraints  of  that  framework,  as  tested  by  the 
principle,  also  Newtonian  in  its  origin,  of  virtual  work.  This 
became  in  time  the  Principle  of  d'Alembert  (1742),  who  did  not 
invent  it,  but  exhibited  its  power  and  developed  its  method  by 
applying  it  to  a  great  dynamical  problem  of  unrestricted  form, 
that  of  the  precession  of  the  equinoxes.  As  a  preliminary  to  its 
solution  he  had  to  develop  in  general  terms  the  equations  of  static 
equilibrium  of  a  system  of  forces  considered  as  applied  to  a  single 
rigid  body  such  as  the  Earth,  that  is,  to  create  a  formal  science  of 
Statics:  and  it  may  be  said  to  be  the  mode  of  development  rather 
than  the  principle  itself  that  constitutes  his  essential  contribution 
to  general  dynamical  theory.  Cf.  the  historical  introductions  in 
Lagrange's  Mecanique  Analytique. 

2.  The  principle  of  the  relativity  of  force  has  recently  become 
prominent  again,  and  pushes  along  further  on  the  same  lines;  it 
now  even  puts  the  question — Are  there  intrinsic  forces  of  nature 
at  all?  May  not  all  force,  including  universal  gravitation,  be  ex- 
pressible as  reaction  against  acceleration  of  motion,  just  after  the 
manner  of  the  obviously  unreal  centrifugal?  On  such  a  view, 
wherever  there  is  a  force  of  gravitation  in  evidence,  its  presence 
must  be  replaced  by  an  acceleration  common  to  all  of  the  material 
bodies  at  each  place  and  relative  to  our  frame  of  measurement, 
of  amount  equal  and  opposite  to  the  intensity  of  the  force.  That 
would  be  the  end  of  the  matter,  if  any  frame  of  reference  could 
be  found  to  satisfy  this  condition.  There  being  then  no  forces  left, 
the  Principle  of  Least  Action  would  make  orbits  simply  the  shortest 
paths  in  the  frame.    Newtonian  uniform  space  and  time  certainly 


326  Sir  Joseph  Larmor 

could  not  permit  this  transformation:  nor  could  the  fourfold 
uniform  continuum  of  interlaced  space  and  time  of  the  earlier 
relativity  theory  be  adapted  to  it.  Will  such  a  fourfold,  deformed 
into  a  non-uniform  and  therefore  non-flat  heterogeneous  space, 
permit  it?  This  is  the  problem  raised  by  Einstein's  idea  of  the 
relativity  of  gravitational  force.  Perhaps  it  goes  even  further,  and 
asks  whether  if  this  will  not  do,  there  can  be  some  other  corpus 
of  abstract  differential  relations  invented,  that  will  transcend 
the  notion  of  spacial  continuity  altogether  but  will  in  compen- 
sation for  that  formidable  complexity  succeed  in  effecting  this 
object. 

In  any  case  we  may  recognise  that  this  merging  of  all  the  forces 
of  nature  into  spacial  relations  satisfies  one  requirement  which  is 
not  quite  the  claim  that  is  explicitly  made  for  it.  The  question 
is  immediately  insistent;  why  should  intrinsic  forces  be  measurable 
with  Newton  in  terms  of  second  gradients  of  type  (Ps/dt^  and  not 
by  a  more  complex  formula  involving  others  as  well?  The  answer 
supplied  by  the  theory  would  be  that  the  idea  of  the  curvature  of 
a  deranged  space  is  expressed  by  a  measure  which  does  not  involve 
higher  gradients. 

It  is  interesting  to  reflect  nowadays  that  in  referring  to  the 
doctrines  of  action  at  a  distance  in  the  preface  to  the  Electricity 
and  Magnetism,  in  1873  Maxwell  classifies  them  as  "the  method 
which  I  have  called  the  German  one,"  and  that  notwithstanding 
Helmholtz's  very  powerful  critical  work  on  Maxwell's  theory,  be- 
ginning in  1870,  that  description  remained  substantially  true  until 
after  Maxwell's  death  in  1879.  Though  he  lived  for  nine  years 
longer  he  seems  to  have  taken  no  part  in  these  discussions  with 
exception  of  a  reference  to  Helmholtz  in  connexion  with  Weber's 
theory  {Treatise,  §  254),  but  worked  chiefly  at  the  development  of 
the  theory  of  stresses  in  gases  regarded  as  molecular  media,  and 
so  in  some  respects  parallel  to  his  theory  of  an  electric  medium. 
He  seems  to  have  been  content  to  leave  his  electric  scheme  to 
germinate  and  expand  in  the  fulness  of  time.  In  connexion  with 
the  recent  efliorts  to  transcend  both  action  at  a  distance  and  an 
aethereal  medium,  his  explanations,  in  an  Appendix  to  the  Memoir 
on  the  determination  of  the  ratio  of  the  electric  units,  Phil.  Trans. 
1868  and  the  critical  chapter  on  '  Theories  of  Action  at  a  Distance' 
in  the  Treatise,  §§  846 — 866,  are  far  from  being  obsolete. 

This  hypothesis  as  to  gravitation,  which  asserts  that  it  is 
essentially  of  the  same  nature  as  the  apparent  increase  of  weight 
which  is  experienced  by  an  observer  going  up  in  a  lift  with  ac- 
celerated motion,  naturally  involves  many  consequences,  and 
raises  questions  regarding  the  relation  of  gravitation  to  physical 
agencies  such  as  light,  the  answer  to  which  may  be  ambiguous  until 
yet  further  postulates  intervene. 


Gravitation  and  Light  327 

Thus  in  the  preliminary  stage  it  occurred  to  Einstein  that  the 
period  of  a  train  of  light  waves  would  be  no  longer  uniform 
throughout  its  course.  Let  us  consider  a  mass  of  hydrogen  gas 
at  P,  say  in  the  Sun,  sending  light- waves  to  an  observer  Q,  both 
being  situated  in  a  region  in  which  there  is  a  field  of  gravitation 
of  intensity  represented  by  </,  directed  from  Q  to  P.  In  terms  of 
the  postulate  of  the  relativity  of  that  force  this  statement  would 
mean  that  the  spacial  frame  to  which  the  underlying  events  are 
referred  is  rushing  as  a  whole  from  P  toward  Q  with  acceleration  g. 
Let  V  be  the  velocity  of  the  frame  at  the  instant  when  a  specified 
light- wave  passes  any  intermediate  point  Q' :  by  the  time  this 
wave  has  reached  Q  the  velocity  of  the  frame  as  a  whole  has  risen 
to  V  -\-  g.Q'Qjc  approximately,  where  g  is  mean  intensity  along 
the  range  from  Q'  to  Q.  Thus  to  the  accelerated  observers  the 
waves  emitted  become  longer  with  distance  traversed,  in  the  ratio 
^  +  9 -Q  Q/c^,  owing  to  this  velocity  of  recession  from  the  source : 
that  is,  the  apparent  wave-length  undergoes  change  so  that 
during  the  progress  from  Q'  to  Q  it  is  altered  in  the  ratio  1  —  SF/c^, 
where  87  is  the  rise  of  potential  (or  fall  of  gravitational  potential 
energy)  along  that  path. 

The  period  of  the  light  will  thus  appear  to  be  increased  to 
different  observers  on  the  line  PQ,  all  of  them  travelHng  along 
with  the  same  acceleration  g,  in  different  degrees  according  to  their 
positions.  This  is  what  will  happen  if  the  observers  and  their  space 
and  optical  instruments  form  a  world  of  their  own  rushing  past, 
or  through,  an  underlying  actual  world,  with  this  acceleration  g, 
instead  of  the  actual  world  rushing  past  them  with  the  opposite 
acceleration  produced  by  a  force  of  gravitation.  For  these  alter- 
natives are  not  now  the  same:  the  finite  velocity  of  propagation  c 
is  constant  with  respect  to  the  actual  underlying  world,  not  the 
observers'  moving  space.  If  the  radiating  hydrogen  belongs  to  the 
actual  underlying  world,  and  the  spectroscopes  of  the  observers 
belong  to  their  own  spacial  scheme  that  is  imposed  on  that  world, 
this  description  is  complete:  the  period  of  each  wave  as  apparent 
to  observers  along  its  path  will  increase  as  the  wave  travels  away 
to  places  of  lower  gravitational  potential.  The  spectral  lines  of 
solar  hydrogen  as  observed  on  the  earth  ought  to  be  displaced 
towards  the  red,  by  the  amount  corresponding  to  the  total  fall 
of  potential  between  Sun  and  Earth.  But  the  postulate  of  two 
worlds  seems  to  be  here  necessarily  involved.  Which  of  them  would 
a  mass  of  radiating  hydrogen  situated  half-way  to  the  Sun  belong 
to?*  The  larger  Doppler-Fizeau  effect  due  to  the  motion  of  the 
source  itself  relative  to  the  observers'  frame  has  not  here  been 

*  All  the  bodies  in  the  space,  being  subject  to  the  same  gravitation,  would 
move  along  with  it:  the  waves  of  light  alone  would  seem  to  be  regarded  as  inde- 
pendent: yet  they  have  energy  and  so  inertia. 


328  Sir  Joseph  Larmor 

mentioned:  that  is  included  satisfactorily  in  tlie  earlier  uniform 
relativity  formulation. 

This  relation  of  light  to  gravitation  is  thus  one  of  the  questions 
raised  by  the  postulate  of  the  relativity  of  that  universal  force. 
Einstein  answered  in  1911*  in  one  way,  that  the  spectrum  of  solar 
hydrogen,  when  compared  with  terrestrial  hvdrogen  which  is  con- 
nected with  the  observer,  should  be  displaced  slightly  towards  the 
red:  but  it  is  a  question  whether  the  consistent  development  of 
that  train  of  ideas  would  not  rather  require  that  it  be  not  displaced 
at  all. 

In  connexion  with  his  later  formal  theory  of  gravitation  the 
same  effect  is  described  as  due  to  varying  local  scales  of  time, 
which  seem  to  be  carried  without  change,  by  the  pulsations  of  the 
rays,  from  the  place  of  their  origin  to  all  the  other  parts  of  the 
universe:  whereas  in  the  above  the  apparent  period f  changes  as 
the  ray  advances.  The  observers  along  the  ray  are  supposed  to 
be  in  communication  with  one  another.  In  so  far  as  their  space 
moves  forward  as  a  whole  it  is  not  stretched  or  shrunk:  in  that 
case  it  can  be  only  their  scales  of  apparent  duration  of  time  that 
are  lengthened  localh'^  by  a  factor,  the  inverse  of  1  —  V jc^.  This 
involves  that  the  scale  of  apparent  velocity  in  the  unchanged  space 
will  be  altered  in  the  direct  ratio:  and  rays  of  light  in  a  field  of 
varying  potential,  if  they  were  paths  of  stationary  time,  might  be 
thought  to  be  deflected.  But  fundamentally  the  path  of  the  ray 
is  determined  by  the  number  of  wave-lengths  in  its  course  being 
made  stationary,  as  compared  with  neighbouring  courses:  and  this 
is,  in  the  present  case,  not  the  same  as  minimum  time  of  transit, 
for  apparent  time  has  lost  its  uniform  scale  while  space  has  not. 

Thus  the  path  of  a  ray  would  be  determined  by  the  condition 
that  SSs/A  summed  along  it  shall  be  stationary:  but  if  there  is 
correspondence  between  the  two  systems  of  reference  which 
changes  all  lengths  around  each  point  in  the  same  ratio  then  hsjX 
will  be  everywhere  the  same  in  both  systems.  The  circumstances 
of  the  path  would  thus  not  be  altered  by  this  change  of  view 
regarding  gravitation,  and  there  ought  to  be  no  special  deviation 
of  the  rays  involved  in  it. 

But  if  g  is  not  uniform  along  the  path  r  of  the  ray,  is  a 
shrinkage  of  the  accelerated  apparent  space  involved?    The  answer 

*  His  exposition  which  has  here  been  paraphrased  is  in  Ann.  der  Physik,  35, 
1911,  §3,  p.  904. 

The  argument  of  this  and  the  next  two  paragraphs  is  based  on  the  implication 
that  in  a  theory  of  transmission  by  contact,  radiation  like  other  things,  the  so- 
called  clocks  included,  must  conform  to  local  measure:  the  alternative,  described 
at  the  end  of  the  paper,- that  racUation  is  extraneous  in  so  far  as  it  imposes  an 
absolute  scale  of  space-time  of  its  own  on  the  whole  cosmos,  was  here  taken  to  be 
excluded  in  advance  from  this  type  of  theory. 

•j-  Measured  on  a  fundamental  scale. 


Gravitation  and  Light  329 

is  given  that,  passing  to  the  general  problem,  the  demands  of  the 
universal  gravitational  correspondence  (to  be  evolved  immediately, 
infra)  require  that  the  apparent  space  of  the  observers  must  be 
constructed  so  that  S/^  —  c"^ht^  where  c  is  a  function  of  r  shall  be 
invariant.  This  requires  slight  warping  of  the  fourfold  space,  so 
that  the  section  in  the  plane  r,  t  is  curved  away  from  its  tangent 
plane.  But  is  the  warped  element  of  extension  ^r'  .c'ht  thereby 
altered  only  to  the  second  order  from  its  corresponding  previous 
normal  value  Sr.cS^?  If  that  be  so,  the  scale  of  t  must  be  altered 
in  the  inverse  ratio  to  the  scale  of  velocity  c'  or  (what  is  the  same 
in  another  aspect)  of  time  t :  and  in  fact  it  is  partly  this  secondary 
change  of  scale  of  r  that  modifies  the  astronomical  gravitation,  as 
will  presently  appear. 

The  answer  to  this  question  might  at  first  be  imagined  to  be  as 
follows :  any  change  in  the  element  of  surface  may  be  made  in  two 
stages,  a  stretching  on  the  original  plane  and  a  displacement  along 
the  direction  normal  to  that  tangent  plane:  it  is  only  the  former 
that  can  produce  a  first-order  effect:  but  this  is  only  an  apparent 
change,  a  mere  alteration  of  coordinates,  because  in  it  the  curvature 
of  the  plane  is  conserved,  so  it  cannot  affect  the  concatenation  of 
relations  or  events  which  alone  counts :  the  latter  does  affect  them, 
e.g.  disturb  the  law  of  gravitation,  but  only  to  the  second  order. 

But  as  will  appear  presently  this  relation  of  conservation  of 
extent  is  between  coordinate  systems  that  most  closely  correspond, 
so  is  a  real  imposed  condition  which  cannot  be  adjusted  by 
change  to  another  set  in  the  fiat.  It  is  the  expression  of,  or  at  any 
rate  is  involved  in,  a  restriction  that  in  the  containing  fivefold 
the  distance  between  corresponding  points  on  the  two  systems  is 
everywhere  small,  so  that  approximate  methods  can  apply  con- 
sistently throughout,  of  which  otherv/ise,  in  making  continuations 
in  an  uncharted  extension,  there  would  be  no  guarantee. 

3.  Now  let  us  survey  this  problem  of  transcending  gravitation 
from  the  other  side,  on  which  it  originated.  With  Minkowski  the 
very  incomplete  relativity  of  electrodynamics,  referring  only  to 
uniform  translatory  convection,  crystallised  into  the  complete  pro- 
position that  events  occur  in  a  uniform  fourfold  of  mixed  space 
and  time,  determined  by  the  consstitutive  spacial  equation 

Here  c  has  nothing  to  do  with  the  velocity  of  radiation :  it  is  simply 
the  dimensional  factor,  prescribing  a  scale  of  measurement,  that 
is  needed  to  make  time  homogeneous  with  length  and  may  be 
taken  as  unity.  Gravitation  remains  outside  this  electrodynamic 
scheme,  being  formulated  in  the  different  Newtonian  reckonings  of 
space  and  time.  Can  it  be  forced  in,  either  exactly  or  approxi- 
mately? 


dt  =  0 


330  Sir  Joseph  Larmor 

The  complete  circumstances  of  the  orbits  in  a  field  of  force  of 
potential  energy  —V  per  unit  mass  (in  a  gravitational  field  V  is 
TiSm/r)  are  condensed  into  the  single  variational  Least  Action 
equation  of  Lagrange-Hamilton, 

with  integration  between  limits  of  time  fixed  and  unvaried.  This 
suggests  comparison  with  the  equation  for  the  shortest  or  most 
direct  path  in  a  modified  fourfold  involving  Euclidean  space  com- 
bined with  a  measure  of  time  varying  from  place  to  place:  for 
that  equation  is 

Sjda=0    where    Sa^  ^  Sx'- +  Sy^  +  Sz^  -  c'^Bt^ 

in  which  c'  is  a  function  of  x,  y,  z.   Let  us  write 

C'2  =  C2  (1    -f  K), 

where  K _is  very  small  on  account  of  the  greatness  of  c.  The 
equation  is  now 

or  approximately  up  to  the  fourth  order 


■&--i---im-m-m. 


dt^  0. 


The  time-limits  being  unvaried  the  first  term  —  c^  can  be  omitted : 
thus  this  variational  equation  of  most  direct  path  coincides  with 
the  previous  orbital  equation  if 

-  |Zc2  =  F. 

Thus  the  forces  are  absorbed  into  a  varying  scale  of  time;  and  the 
motion  being  now  free  under  no  force,  the  orbit  is,  as  was  antici- 
pated, a  geodesic  or  straightest  path.  The  orbits  have  become 
however  straightest  paths,  not  in  their  original  Newtonian  separ- 
ated space  and  time,  but  in  the  uniform  space-time  fourfold  of 
relativity  as  slightly  deranged  by  the  not  quite  constant  scale  of 
time. 

Thus  the  orbits  in  any  field  of  attraction  have  actually  been 
fitted  into  the  mixed  space- time  frame  of  electrodynamic  relativity, 
at  the  expense  of  doing  slight  violence  to  that  frame,  by  making 
the  measure  of  time  vary  from  place  to  place  while  the  positional 
specification  remains  uniform. 

But  this  transformation  does  more  than  is  needed.  It  ought 
somehow  to  be  restricted  to  the  one  universal  force  of  nature,  that 
of  gravitation  with  its  inverse-square  law.  It  is  here  that  the 
special   feature   of  the  Einstein  theory  seems  to  come  in.    For 


Gravitation  and  Light  331 

velocities  beyond  actual  astronomical  experience,  not  small  com- 
pared with  that  of  light,  mass  comes  to  depend  on  speed;  thus  it 
is  not  any  longer  available  as  a  definite  dynamical  constant.  On 
the  earlier  uniform  relativity  it  emerged  however  definitely  in 
another  way  as  a  feature  of  every  permanent  collocation  of  energy 
and  proportional  to  its  amount  E,  equal  in  fact  to  Ejc^.  This 
follows  immediately  if  Least  Action  is  fundamental.  Thus  it  is 
grouped  energy  that  possesses  located  momentum:  and  it  is  this 
energy  that  has  to  gravitate,  mass  confined  to  matter  alone  having 
proved  inadequate  to  a  Least  Action  formulation  in  the  mixed 
space-time  of  universal  limited  relativity.  Dynamical  principles 
had  therefore  to  take  the  form  of  a  theory  of  conservation  of  energy 
and  of  abstract  momentum  as  they  travel  through  a  medium,  at 
the  same  time  receiving  additions  by  the  operation  of  an  internal 
stress  to  which  the  medium  is  to  be  subject.  In  other  words, 
general  dynamics  cannot  be  more  detailed  than  a  mere  description 
of  the  migration  of  energy  and  of  momentum  in  a  medium  under 
the  influence  of  some  internal  system  of  stress  adjusted  to  fit  the 
equations  as  simply  as  possible.  This  stress  is  what  has  to  stand 
for  or  represent  the  agencies  of  nature.  The  theory  is  borrowed 
and  generalised  from  the  Maxwellian  theory  of  stress  in  the  aether, 
which  was  an  isolated,  apparently  rather  accidental,  feature  that 
did  not  fit  well  into  the  substance  of  Maxwell's  scheme,  because  in 
fact  it  could  not  be  connected  with  a  strain  expressive  of  its 
origin.  Now  however,  inertia  of  bodies  having  failed  as  the  standard 
measure  of  force,  energy  and  momentum,  and  a  postulated  ad- 
justing stress  entirely  at  our  choice,  are  promoted  to  occupy  the 
vacant  place.  Only  it  is  not  called  a  stress:  the  idea  of  a  physical 
medium  is  avoided,  so  it  is  named  an  algebraic  tensor.  There  is 
no  law  of  elasticity  involved,  or  relation  of  stress  to  strain,  such 
as  makes  elastic  problems  determinate.  Thus  the  scheme  may 
have  accidental  features,  is  perhaps  far  from  being  unique.  Another 
parallel  to  it  is  Maxwell's  theory  of  stresses  in  a  gas  due  to  varying 
temperature:  but  that  continuous  theory  could  never  have  been 
constructed  in  definite  form  without  the  foundation  of  the  be- 
haviour of  the  individual  molecules. 

When  however  the  fourfold  frame  is  very  nearly  flat,  the  rela- 
tions of  energy-momentum-stress  appear  to  fall  in  with  the  law  of 
gravitation,  with  energy  as  the  source  of  its  potential  instead  of 
matter. 

When  the  deranged  spacial  frame  nowhere  differs  much  from 
the  flat,  it  may  be  expected  that  the  extent  of  its  fourfold  element 
will  be  altered  from  the  value  for  coordinates  of  the  corresponding 
type  on  the  flat  only  to  the  second  order,  for  the  same  kind  of 
reason  as  applies  in  comparing  a  slightly  deranged  plane  sheet  with 
the  original  plane.  In  fact,  if  the  displacement  is  everywhere  small, 


332  Sir  Joseph  Larmor 

this  extent  taken  over  a  small  region  would  have  a  stationary 
value  for  the  flat,  changing  in  the  same  direction  on  both  sides 
of  it.  Cf.  supra,  p.  329.  Thus  for  a  spherically  symmetrical  field 
the  constitution  of  the  fourfold  must  be  determined  in  polar 
coordinates  by  the  equation 

Sct2  -  {c/c'f  8/2  +  {rSdf  +  (r  sin  O^f  -  c'^Si, 

showing  that  the  positional  part  of  the  extension  is  very  slightly 
non-uniform  and  so  not  quite  Euclidean.  It  appears  to  be  this 
secondary  feature,  not  the  energy-momentum-stress  tensor  con- 
ditions, that  modifies  gravitation  from  the  Newtonian  law. 

The  expositions  of  relativity  do  not  mention  an  extended 
fourfold,  which  would  be  foreign  to  the  cardinal  idea  that  space 
is  constructed  from  physical  origins,  only  in  so  far  as  it  is  needed — 
even  though  it  has  to  be  implied  that  it  is  reproduced  unerringly 
each  time.  But  the  instrument  of  such  construction  or  continua- 
tion of  a  metric  space  is  an  infinitesimal  linear  measuring  rod 
supposed  to  have  complete  free  mobility  without  change  of  in- 
trinsic length :  and  it  would  seem  to  be  a  tenable  view  that  such 
a  mobile  apparatus  must  determine  an  underlying  flat  space  of 
higher  dimensions*  in  which  the  physical  system  may  be  supposed 
imbedded. 

It  is  to  be  noted  here  that  a  surface  defined  intrinsically  in  the 
Gaussian  manner  by  the  distance  relation  on  it 

Ss^  =fSp^  +  2gSpSq  +  hSq^, 

remains  the  same  surface  when  the  coordinate  quantities  p,  q  are 
changed  to  others  p  ,  q'  which  are  any  assigned  functions  of  them 
both,  so  that 

Ss^=.f'8p'-^  +  2g'8p'8q' +  h'8q'^, 

provided  8s  is  measured  by  the  same  infinitesimal  unchanging 
measuring  rod  extraneous  to  the  surface  in  both  cases.  These  two 
equations  represent  the  same  surface,  only  the  generalised  co- 
ordinates of  the  same  point  on  it  are  changed  from  {p,  q)  to  {p,  q'). 
The  intrinsic  curvatures  are  the  same  from  whichever  form  they 
be  calculated:  if  one  form  represents  a  flat,  so  does  the  other.  On 
this  definition  by  an  intrinsic  differential  relation  surfaces  are 
indistinguishable,  if  one  can  be  bent  to  fit  the  other  without 
stretching.  So  in  the  Riemann  theory  of  spaces  of  more  than  two 
dimensions  it  is  the  functional  forms  of  the  coefficients  in  the 
quadratic  function  of  differentials  and  the  mobile  absolute  mea- 
suring rod  that  determine  the  nature  of  the  space;  any  transforma- 
tion of  coordinates  changes  the  coefficients  (or  potentials  in  the 
gravitational   formulation)    but   so   that   the   space   remains   un- 

*  For  a  radial  field  it  need  be  of  onlj'  one  more  dimension. 


Gravitation  and  Light  333 

changed,  being  only  referred  as  regards  the  same  points  to  the 
other  generalised  coordinates.  But  the  apparent  extent  ^pdq  does 
alter  when  the  coordinates  are  changed,  and  it  would  be  a  limita- 
tion to  keep  it  constant.   See  Appendix  infra. 

The  feature  that  remains  unfathomed  as  yet  is  the  fact  that 
the  velocity  of  transfer  of  energy  of  radiation  in  undisturbed  regions 
of  space  is  equal  to  the  merely  dimensional  constant  that  renders 
time  comparable  with  space  on  the  fourfold  frame  of  reference :  it 
at  any  rate  suggests  a  dynamical  origin  for  that  mixture  of  the 
effective  relations  of  time  with  those  of  space  *. 

The  locus  in  the  fourfold  in  which  a  never  changes  and  so  ha 
vanishes  has  some  claim  to  be  called  the  'absolute,'  in  a  sense 
parallel  to  the  '  absolute '  of  Cayleyan  geometry  which  for  Euclidean 
space  is  represented  by  the  equation  x^  +  y^  +  z^  =  0.  Everywhere 
on  this  locus  S.s  =  c'ht ;  thus  velocity  of  displacement  is  everywhere 
c',  and  the  rays  in  it  are  the  paths  of  shortest  time  with  this 
velocity.  It  separates  the  disparate  regions  in  which  ha  measures 
real  distance  when  time  is  unvaried  and  in  which  iSct  measures  real 
time  when  position  is  unvaried. 

4.  It  would  appear  (as  infra,  p.  335)  that  if  we  are  prepared  to 
replace  a  field  of  potential  energy  of  gravitation  or  any  other  type  of 
universal  force  by  a  field  of  varying  time-scale  without  change  of  the 
uniform  scale  of  space,  on  the  lines  sketched  above,  this  formal 
change  ought  not  sensibly  to  affect  radiation  either  as  regards  its 
path  or  its  period.  To  each  element  of  extent  there  would  be  a  cor- 
responding element,  and  all  events  and  measures  in  one  pass  over  to 
the  other  according  to  rule. 

But  we  now  pass  from  kinematic  discussion  of  frames  of  refer- 
ence to  physical  considerations.  If  we  are  to  assert,  in  agreement 
with  the  doctrine  of  relativity  plus  Least  Action,  that  inertia  is  a 
property  of  organised  energy  and  proportional  to  it,  therefore  not 
solely  of  matter,  and  if  we  are  to  admit  with  Einstein,  in  the  same 
and  other  connexions,  that  light  is  made  up  of  small  discrete 
bundles  or  quanta  of  energy,  it  would  appear  to  follow  that  each 
bundle  is  subject  to  gravitation.  Therefore  if  a  bundle  comes  on 
from  infinite  distance  with  velocity  c,  when  it  has  reached  a 
place  of  potential  V  near  the  Sun  its  velocity  c  must  be  given  by 

ic'2  -  F  -  ic2, 

in  other  words,  is  increased  in  the  ratio  1  +  V jc^.  It  will  swing 
round  the  Sun  in  a  concave  hyperbolic  orbit,  and  as  the  result, 
the  direction  of  its  motion  will  suffer  deflection  away  from  the  Sun 
by  half  the  amount  that  has  been  astronomically  observed. 

This  reasoning  would  not  be  estopped  by  the  principle  that  c  is 
the  upper  limit  of  possible  material  velocities:  for  that  is  because 

*  See  final  paragraphs. 


334  Sir  Joseph  Larmor 

a  moving  body  acquires  energy  and  therefore  inertia  without  hmit 
as  its  speed  approaches  c,  whereas  the  energy  of  a  Hght  quantum 
is  not  supposed  so  to  increase. 

This  is  all  on  the  older  notions:  the  velocity  c  is  far  too  great 
for  the  new  approximate  gravitation  analysis  to  be  applicable. 
But  the  idea  of  wavefronts  and  phases  must  also  be  introduced 
somehow.  If  we  imagine  a  row  of  these  corpuscles  of  energy  coming 
on  abreast,  the  more  distant  ones  would  fall  behind  in  swinging 
round  the  Sun  and  their  common  front  would  become  oblique  to 
their  direction  of  motion,  the  exactly  transverse  directions  being 
now  the  loci  of  equal  Action  not  of  equal  time.  If  we  superposed 
the  Huygenian  principle  of  propagation  normal  to  the  front,  the 
orbital  deflection  would  thereby  be  just  cancelled  by  the  swinging 
back  of  the  front  which  would  retain  its  direction :  and  there  would 
be  no  deflection  of  direction  of  propagation.  But  such  ideas  are 
plainly  incoherent. 

The  earlier  development  of  Einstein  sketched  above*  was 
driven  on  other  grounds  to  conclude  that  light  must  gain  energy 
in  a  field  of  gravitation,  but  the  gain  was  named  potential  energy. 
In  the  finally  developed  theory  there  seems  to  be  no  longer  energy 
of  motion  or  other  types :  energy  becomes  a  single  analytic  scalar 
in  what  is  left  of  the  field  of  interplay  of  momentum,  energy  and 
stress. 

These  earlier  considerations  have  doubtless  crystallized  into 
the  formal  theory  of  which  also  the  result  has  been  illustrated 
above,  in  a  way  which  transforms  the  variational  equation  of  free 
orbits  in  ordinary  space  and  time  into  the  variational  equation  of 
straightest  lines  in  a  non-uniform  space-time  fourfold  given  differen- 
tially. The  coordinates  are  carried  over  unchanged  in  values,  into 
this  fourfold,  but  their  differentials  no  longer  express  in  it  direct 
measurements  of  length  and  time;  these  are  now  imported  in  the 
Riemann  manner  as  regards  any  element  of  arc  or  interval  of 
time  by  the  value  of  the  absolute  element  8a.  As  compared  with 
the  underlying  absolute  time  determined  by  Scr,  the  element  of 
apparent  time  St  of  a  gravitational  world,  which  is  taken  over  into 
its  expression  is  variable,  proportional  to  c'"^,  with  locality. 

The  quantities  x,  y,  z,  t  which  are  the  measures  of  space  and 
time  as  apparent  in  the  world  of  gravitation  are  now  mere  co- 
ordinate quantities  in  the  new  differentially  given  world  in  which 
there  are  elements  of  absolute  length  and  time  both  measured  by 
Sct.   The  final  expression  for  Scr^  with  radial  symmetry 

,f  Sr2+  ...  +  ...  -  (^y^^S^^ 
shows  that  the  element  of  apparent  time  in  the  gravitational  world 

*  Ann.  der  Physik,  35,  1911,  §2,  p.  902. 


Gravitation  and  Light  335 

is  the  unchanging  element  of  absolute  time  divided  by  c' Ic,  or  that 
the  scale  of  apparent  time  is  variable  with  locality  in  the  ratio  cjc' : 
also  that  the  scale  of  apparent  radial  length  is  variable  in  the 
ratio  c  jc:  and  therefore  the  scale  of  radial  velocity  is  variable  as 
their  quotient  c^jc"^.  How  then  with  respect  to  the  velocity  of 
rays  of  light  whose  absolute  value  is  the  same  as  the  dimensional 
constant  c?  Referred  to  these  variable  scales  its  apparent  value 
along  any  element  of  arc  ought  to  be  changed  at  the  same  rate 
as  any  other  velocity  along  that  element  of  arc  would  be  changed, 
if  rays  are  not  to  remain  outside  the  correspondence  between 
hx,  8y,  Sz,  St  representing  time-space  in  the  apparent  gravitational 
world  and  the  same  quantities,  now  elements  of  mere  coordinates  in 
a  difEerentially  given  world  in  a  curved  space-time  which  has 
absorbed  gravitation.  This  maintenance  of  correspondence  is 
secured  if  we  determine  the  ray- velocity  along  any  element  of  arc 
by  making  Scr  =  0 :  and  the  modified  theory  of  radiation  for  the 
apparent  space  of  gravitation  must  be  such  as  can  accept  this 
value  of  the  velocity  of  propagation  *.  The  correspondence  takes 
over  the  same  values  of  the  coordinate  differential  elements.  In 
the  apparent  gravitational  world  they  represent  its  space  and  time, 
in  the  new  world  differentially  specified,  they  belong  to  mere 
coordinates:  absolute  elements  of  space  and  of  time  are  there  ex- 
pressed by  8o-,  but  a  relation  of  scales  can  be  established  from  the 
formula  which  expresses  Sct^. 

The  transformation  which  changes  orbits  into  geodesies  in  the 
difEerentially  given  space-time  does  not  turn  rays  into  rays:  their 
velocity  is  too  great  and  moreover  their  minimum  property  is 
relative  to  their  locus  8a  =  0.  But  if  the  ray  is  supposed  to  have 
a  constant  underlying  absolute  period  of  pulsation  and  a  constant 
absolute  wave-length  (and  therefore  to  be  a  straight  line  in  an 
auxiliary  uniform  fivefold)  its  apparent  period  in  the  gra^dtational 
world  must  vary  with  locality  as  (c'/c)"^,  also  its  apparent  element 
of  length  inversely  as  the  scale  of  length  pertaining  to  its  direction 
on  that  locality,  and  its  apparent  velocity  as  before  specified.  Its 
apparent  path  in  the  gravitational  world  will  correspond  to  the 
true  absolute  path  Sfda/X^  ==  0,  therefore  will  be  given  by 

Sjds/X  =  0, 

complications  being  avoided  as  fortunately  t  is  not  involved  ex- 
plicitly in  these  equations.  But  at  the  same  place  the  scales  of 
apparent  8s  and  apparent  A  would  alter  on  the  same  ratio  owing 
to  the  presence  of  gravitation :  therefore  its  influence  is  eliminated 
in  the  quotient,  and  the  path  is  not  affected  by  the  gravitation, 
is  the  same  whatever  be  its  intensity.  A  ray  passing  near  the  Sun 
ought  not  to  be  deflected  on  this  view:  an  observed  deflection, 

*  On  this  and  the  following  paragraphs,  cf.  however  the  end  of  the  paper. 
VOL.   XIX.   PART  VI.  23 


336  Sir  Joseph  Lannor 

whicli  a  priori  was  well  worth  looking  for,  would  seem  to  await 
explanation  on  other  lines. 

Again  would  there  be  an  observable  change  of  periods  of 
spectral  lines  according  as  the  vibrating  source  was  at  the  Sun  or 
at  the  Earth?  The  underlying  absolute  periods  of  radiating 
hydrogen  molecules  would  be  always  and  everywhere  the  same: 
thus  the  apparent  period  in  the  gravitational  world  would  vary 
inversely  as  the  local  scale  of  time,  and  be  longer  at  the  Sun. 
But  this  is  a  local  apparent  period.  The  waves  sent  out  from  the 
solar  molecule  are  observed  at  the  earth:  we  have  seen  that  their 
length  changes  as  they  progress,  being  inversely  as  the  local  scale 
of  length,  and  their  speed  changes  also,  so  that  their  period  changes 
inversely  as  the  local  scale  of  time.  Thus  when  they  have  reached 
the  Earth  their  period  conforms  to  the  local  scale  and  would  agree 
with  that  of  the  radiation  of  a  similar  terrestrial  molecule.  In  fact 
if  complete  correspondence  is  established*,  element  for  element, 
as  above,  all  periods  or  intervals  of  time  measured  at  any  element 
are  changed  in  the  same  ratio  depending  on  the  locality  alone. 
Any  other  conclusion  would  make  the  pulsating  rays  into  signals 
establishing  absolute  time  throughout  the  apparent  universe, 
which  could  hardly  be  a  result  of  a  theory  of  relativity. 

The  condition  8ct  =  0  prescribes  a  definite  ray- velocity  for  each 
element  of  arc,  the  same  forwards  as  backwards,  only  when  Scr^ 
involves  St'^  but  no  products  of  St  with  other  differentials:  in 
other  cases  it  gives  two  velocities,  not  equal  and  opposite,  and 
this  spacial  scheme  of  rays  seems  to  fail.  If  rays  are  to  be  pro- 
perties of  the  space  a  very  severe  restriction  is  thus  imposed  on 
the  form  of  8cr^,  but  one  which  seems  to  be  satisfied  for  the  slight 
modifications  that  would  be  involved  in  the  actual  gravitation  of 
experience. 

In  the  modifications  of  the  expression  for  8cr^  which  absorb 
gravitation  the  coefficients  do  not  involve  the  time  explicitly: 
therefore  the  ray-paths  are  fixed  in  the  space,  and  it  almost  looks 
as  if  they  were  guides  imposed  by  the  nature  of  the  space  alone, 
as  thus  modified,  for  the  alternating  energies  of  radiation  to  run 
along. 

Any  inference  that  because  a  ray  is  fixed  in  space,  as  many 
waves  must  run  in  at  one  end  as  run  out  at  another,  would  be  at 
variance  with  the  very  notion  of  relativity,  by  providing  a  scale 
of  absolute  time  throughout  the  universe.  Such  an  argument 
seems  to  amount  in  more  general  form  essentially  to  this:  when 
the  expression  for  8a^  does  not  contain  t  explicitly  it  will  make  no 

*  As  has  been  estabKshed  for  the  more  general  case  in  a  beautiful  analysis  by 
Prof.  Th.  de  Bonder,  of  Brussels,  Comjptes  Rendus,  July  6,  1914,  Archives  du 
Musee  Teyler,  Haarlem,  vol.  iii,  1917,  pp.  80-180.  [It  is  merely  continuity  with 
non-gravitational  fields,  and  not  correspondence,  that  is  established.] 


Gravitation  and  Light  337 

difference  to  the  cosmos  if  t  is  everywhere  increased  by  the  same 
constant:  therefore  the  scale  of  time  must  be  everywhere  the  same 
— which  excludes  any  possibility  of  local  scales  of  time,  A  change 
of  origin  of  measurement  for  time  is  not  the  same  as  progress  of 
events  in  time,  unless  the  scale  of  time  is  everywhere  the  same. 

The  matter  may  be  put  from  a  different  angle  as  follows.  To 
obtain  the  time  of  transit  of  a  ray  from  P  to  Q  it  is  not  possible 
to  add  elements  of  heterogeneous  local  times  such  as  8^*.  What 
can  be  done  is  to  find  the  true  underlying  time  of  transit.  If  this 
homogeneous  true  time  is  delayed  at  the  start,  at  one  end  of  the 
path  at  P,  it  is  delayed  by  an  equal  amount  at  arrival  at  the  other 
end,  as  the  equations  of  transit  do  not  involve  this  time  explicitly: 
hence  apparent  times  at  the  two  ends  are  delayed  not  by  equal 
amounts,  but  by  amounts  inversely  as  their  local  scales,  so  that 
a  ray  cannot  (as  has  been  impKed)  transmit  apparent  time  along 
its  path. 

The  alternative  development  is,  as  above,  that  8ct^  being  the 
underlying  unchanging  standard  there  are  local  scales  of  time,  and 
local  scales  of  length  which  may  involve  direction,  and  therefore 
also  of  velocity  (including  that  of  the  rays)  which  is  their  quotient. 
The  path  of  a  ray  from  point  to  point  is  determined  by  making 
the  number  of  wave-lengths  from  the  one  to  the  other  minimum, 
that  is  by  Sjds/X  =  0 :  but  Ss  and  A  are  both  altered  to  the  same 
scale;  thus  there  is  no  alteration  due  to  gravitation  in  the  varia- 
tional equation  determining  the  ray-path,  so  that  it  would  suffer 
no  deflection.  The  essential  feature  in  the  argument  is  that, 
whether  rays  may  be  regarded  as  the  limiting  case  of  free  orbits 
or  not,  their  specification  has  been  postulated  so  that  the  ray- 
velocities  correspond  in  the  same  way  as  all  other  velocities  in 
the  two  frames. 

Appendix. — On  Space  and  Time. 

Let  us  try  for  a  closer  realization  of  these  abstract  positions. 
The  Gauss-E,iemann  theory  for  an  ordinary  curved  surface  will  be 
wide  enough  to  serve  as  an  illustration.  The  theory  involves 
coordinates  p,  q:  they  must  represent  something.  The  very  least 
we  can  do  for  them  is  to  regard  the  surface  as  twofold  extension 
dotted  over  with  points,  so  that  the  coordinates  express  their 
order  of  arrangement  according  to  some  plan  of  counting  them 
with  respect  to  this  extension  in  which  they  lie.  There  is  no  metric 
idea  at  all  in  this  numeration,  and  nothing  to  distinguish  one 
surface  from  another.    Now  bring  in  an  infinitesimal  unchanging 

*  Yet  it  is  just  such  elements  of  quasi-time  d.v^  that  are  added  together,  ■infra 
p.  343.  It  is  the  so-called  shifting  clock-time  and  absolute  time  running  parallel 
that  are  the  source  of  all  this  confusion. 

23—2 


338  Sir  Joseph  Larmor 

measuring  rod,  which  can  make  play  in  each  element  of  extension 
represented  by  SpSq  and  also  be  transferred  from  place  to  place: 
and  we  can  thereby  impart  or  rather  superpose  metric  quality  on 
the  twofold  which  hitherto  was  purely  positional  or  rather  tactical. 
The  simplest  plan  is  to  follow  Euclid,  on  the  basis  of  the  Pytha- 
gorean theorem,  and  expressing  absolute  length  according  to 
measuring  rod  by  a  symbol  Ss,  to  impose  a  scale-relation  of  form 

8s^  =  Sp^  +  Sq^. 

But  this  metric  cannot  be  applied  consistently  over  a  curved 
surface,  unless  it  is  of  the  very  special  type  that  can  be  rolled  out 
flat:  for  other  surfaces  it  is  necessary  to  have  the  more  general 
type  of  relation 

S52  =fSp^  +  ^g^pBq  +  hSq^ 

in  which/,  g,  h  are  functions  of  the  coordinates  p,  q. 

This  specification  of  an  imported  metric  thus  determines  the 
surface:  starting  from  a  given  small  region  of  it,  the  form  of  the 
surface  in  an  outer  threefold  space  can  be  gradually  evolved  by 
prolongation  so  as  to  fit  in  with  consistent  application  of  this 
metric.  It  is  this  idea  of  prolongation  of  a  non-uniform  manifold, 
equivalent  to  its  geometrical  continuation  within  a  flat  one  of 
higher  dimensions,  that  was  Riemann's  contribution  to  the  ideas 
of  geometry.  But  the  manifold  itself  is  supposed  to  be  given  only 
tactically  or  descriptively;  and  it  is  the  metric  that  is  imposed  on 
it  that,  by  its  demand  for  consistency  in  measurements,  deter- 
mines for  it  a  form,  as  located  in  a  higher  flat  manifold.  This  form 
is  expressed  in  detail  analytically  by  the  '  curvature '  at  each  place, 
as  specified  by  a  set  of  functions  (one  in  the  case  of  a  surface)  of 

the  successive  gradients  of  the  set  /,  g,h, If  we  keep  the  system 

self-contained  by  avoiding  the  immersion  of  it  in  a  uniform 
auxiliary  manifold  of  higher  dimensions,  our  resource  is  to  deter- 
mine the  curvature  as  the  simplest  set  of  functions  that  are  invariant 
for  local  changes  of  coordinates.  But,  in  order  of  evolution  at  any 
rate,  this  invariance  may  be  held  to  be  only  a  derived  idea. 

In  any  case  the  nature  of  the  non-uniform  manifold,  as  thus 
determined  by  a  metric  imposed  on  formless  space,  has  nothing 
to  do  essentially  with  the  coordinates  p,  q,  ...  to  which  it  may 
happen  to  be  referred:  it  is  settled  by  the  algebraic  form  of  the 
functions/,  g,  h,  ...  expressed  in  terms  of  jj,  q,  ...,  or  in  geometric 
terms  by  the  'curvature'  as  so  expressed. 

As  a  consequence,  if  we  transform  a  surface  from  internal  or 
intrinsic  coordinates  p,  q,  to  others  p',  q',  which  are  assigned 
functions  of  the  former,  so  that  we  obtain 

§§2  =  f'8p'^  +  2g'8p'8q'  +  h'8q'^ 
and  construct  the  surface  implied  in  this  new  equation  by  the 


4 


Gravitation  and  Light  339 

process  of  continuation,  it  will  prove  to  be  just  the  same  surface 
as  before.  Whether  it  is  expressed  in  terms  of  p' ,  q'  or  of  jp,  q  is 
intrinsically  of  no  consequence :  the  coordinates  are  of  no  account, 
it  is  only  the  functional  forms  of/,  g,  h  that  are  essential. 

This  last  statement,  developed  in  terms  of  the  criterion  of 
invariance  in  order  to  avoid  a  representation  by  immersion  in  a 
uniform  geometrical  manifold  of  dimensions  higher  than  the  given 
four  of  space  and  time,  appears  to  cover  the  general  relativity 
of  Einstein.  The/,  g,  h, ...  can  be  named  the  potentials  which  deter- 
mine the  space.  In  the  special  relativity,  before  gravitation  was 
absorbed  into  the  metric  of  extension,  all  spaces  were  flat,  so 
/,  g,  h,  ...  were  constants ;  which  is  all  that  is  left,  for  that  particular 
case,  of  these  relations  of  invariance. 

In  this  flat  fourfold,  relativity  implied  merely  that  a  physical 
system  is  determined  by  its  own  internal  relations,  so  that  the 
position  that  may  be  assigned  to  it  in  the  fourfold  is  of  no  account, 
any  more  than  is  the  position  of  a  surface  or  a  system  of  bodies 
in  space.  In  the  later  general  relativity  the  manifold  must  be 
supposed  given  descriptively  by  coordinates,  which  represent 
numerical  counts  arranged  to  suit  the  number  of  dimensions  that 
are  involved :  it  only  gains  internal  form  when  a  metric  is  imposed 
upon  it.   If  the  Euchdean  metric 

§s^  =  Sp^  +  Sq^+  ... 

is  imposed  it  becomes  a  Euclidean  space  everywhere  uniform  and 
also  flat,  in  which  bodies  are  mobile  without  change  of  form.  If 
a  metric  varying  with  position  is  imposed,  the  expressions  in  this 
manifold  of  the  metric  relations  of  nature  will  become  complicated, 
and  the  relations  so  changed  be  described  as  a  modified  set  of  laws. 

The  original  non-metric  continuum  might  be  marked  for 
instance  by  gradations  of  colour:  the  colour-scheme  of  Newton  as 
developed  by  Young,  Helmholtz,  and  Maxwell,  is  the  standard 
example  of  a  non-metric  threefold  extension. 

May  we  not  here  have  refined  down  to  the  unresolvable  essence 
of  space,  as  the  mere  possibihty  of  descriptive  continuity  of  three- 
fold type  which  is  an  essential  feature  in  our  mental  world  ?  Within 
this  a  priori  datum  of  threefold  uncharted  pure  continuity  we  may 
construct  types  of  charted  spaces  almost  without  limit,  by  imposing 
metrics  of  various  types.  Any  particular  space  is  not  however 
determined  by  the  system  of  coordinates  of  reference  p,  q,  ...  but  by 
the  variable  coeSicients  f,  g,  h,  ...  of  the  imposed  metric  expressed 
as  functions  of  them.  But  yet  it  is  only  under  special  conditions 
when  it  is  uniform  and  flat  that  finite  difl'erences  of  these  co- 
ordinates can  be  involved,  this  being  part  of  the  expression  of  the 
mobility  of  solid  bodies  in  the  space.  It  is  in  this  narrower  sense, 
that  "the  system  of  coordinates  is  accidental,  that  relativity  has 


340         .  Sir  Joseph  Larmor 

now  expelled  general  metric  ideas  of  position.  Would  it  be  entirely 
wrong  to  assert  that  local  or  sectional  relativity  has  been  retained 
for  nature,  so  far  as  this  order  of  ideas  extends,  by  transferring  the 
laws  of  nature  into  a  space-time  frame  which  itself  no  longer 
possesses  that  quality? 

The  distinction  has  thus  been  made  between  an  ultimate  idea 
of  space  as  mere  threefold  continuity,  marked  but  uncharted,  and 
the  metric  that  may  be  imposed  on  it  by  which  it  becomes  a  frame 
fit  for  the  purposes  of  description  of  nature.  There  is  only  one 
space:  but  its  practical  aspect,  whether  Euclidean  or  elliptic  or 
merely  heterogeneous,  depends  on  the  metric  that  we  choose  to 
assign  to  it.  The  metric  would  thus  appear  to  pertain  more  closely 
to  the  order  of  nature  for  which  it  is  to  form  the  most  convenient 
frame  for  description,  than  to  space  itself.  For  space  is  primarily 
bare  threefold  continuity;  though  a  set  of  descriptive  coordinates 
jp,  q,  ...  is  unavoidable  as  a  foundation  of  thought,  any  set  is  as 
valid  as  any  other.  For  ultimately,  the  count  or  census  of  the  points 
or  marks  that  pervade  the  continuity  and  render  it  descriptively 
given  to  us,  is  the  same  count  however  it  be  made.  May  we  say 
that  the  insistent,  originally  uncritical,  notion  of  relativity  reduces 
itself  ultimately  into  this  postulate,  that  as  nature  is  presented  to 
us,  it  is  such  that  in  mental  operations  we  need  attend  only  to 
one  portion  of  the  spacial  continuity  at  a  time?  This  makes  the 
onefold  time,  or  rather  mere  temporal  succession  as  representable 
by  the  8a  of  Minkowski,  the  fundamental  feature*,  which  however 
diverges  spacially  into  a  manifold:  according  to  Hamilton  long 
ago,  algebra  was  the  science  of  pure  time. 

In  the  above,  space  is  given  by  a  manifold  array  of  points,  of 
which  the  coordinates  p,  q,  ...  express  one  of  the  varieties  of 
numerical  census.  Is  then  space-time  absolute,  or  is  it  continually 
being  constructed  by  physical  science  as  it  ranges  over  the  void, 
for  its  own  purposes,  just  to  the  extent  that  it  may  be  required? 
May  we  say  that  the  formless  manifold  is  the  fundamental  feature, 
that  the  array  of  points  and  their  census  do  not  need  to  be 
definite  in  any  respect  a  priori,  and  that  the  metric  which  is 
imposed  on  it  and  makes  it  into  a  definite  working  type  of  space 
is  related  to  the  physical  world  and  so  is  to  be  regarded  as  evolved 
in  connexion  with  our  organic  description  or  mapping  of  nature, 
and  to  be  just  as  permanent? 

What  remains  of  the  original  notion  of  relativity  after  this 
sifting  of  ideas  would  then  coincide  with  the  principle  of  Newton, 
Faraday  and  Maxwell,  originated  by  Descartes,  that  the  operations 
of  nature  are  elaborated  in  fourfold  extension  according  to  a  scheme 
purely  differential,  that  is  by  transmission  from  element  to  element 

*  The  spacial  sign  here  attached  to  8(r^  is  an  accident  of  the  order  of  exposition. 


J 


Gravitation  and  Light  341 

of  the  cosmos,  in  no  case  leaping  across  intermediate  elements  as 
action  at  a  distance  would  imply.  The  early  stage  of  formulation 
of  the  confused  notion  of  relativity  is  the  postulate  that  position 
and  change  of  position  are  purely  relative:  the  final  solution  is  to 
abolish  the  idea  of  immediate  ^m/e  change  of  position  altogether. 
But  that  does  not  imply  that  a  portion  of  the  cosmos  can  evolve 
itself  without  constant  interference  from  all  the  rest. 

To  a  question  as  to  what  is  gained  by  absorbing  gravitation  in 
space  an  answer  would  be  that  it  need  make  no  difference  as  regards 
gravitation ;  but  if  other  relations  of  an  assumed  space- time  fourfold 
(e.g.  stress-tensor  theory)  have  to  go  in  also  in  a  simple  way,  it 
may  be  convenient  or  even  necessary  to  assist  them  by  choosing 
a  space  which  requires  some  alterations  of  the  recognised  laws  of 
gravitation  and,  if  these  suggested  discrepancies  are  verified,  that 
may  presumably  have  a  claim  to  be  the  real  type  of  space.  The 
aim  is  not  primarily  to  reduce  gravitation  to  a  quality  of  space, — 
perhaps  is  not  even  relativity,  which  has  evaporated, — but  is  to  get 
it  out  of  Newtonian  space  and  time  into  the  mixed  space-time 
fourfold  which  was  strongly  suggested  by  the  form  of  the  Max- 
wellian  electrodynamic  relations  of  free  space,  and  would  make 
that  scheme  valid  for  great  velocities  of  convection  beyond  ex- 
perience, even  up  to  the  speed  of  light. 

An  expansion  of  the  Einstein  ideas  on  general  relativity  has 
been  worked  out  by  H.  Weyl  {Ann.  der  Physik,  59,  1919)  in  which 
a  further  metric  scale  of  vector  character  appears  to  be  imposed 
on  a  non-uniform  space-time,  which  has  here  been  itself  ascribed 
to  the  imposition  of  a  Gauss- Riemann  metric  on  the  formless 
spacial  threefold  that  is  inherent  in  the  mind.  There  would  seem 
to  be  no  formal  obstacles  to  such  piling  up  of  metric  upon  metric, 
in  an  unlimited  play  of  thought. 

The  physical  analysis  perhaps  not  very  remote  to  this  new 
elaboration  of  metric  is,  as  I  think  Prof.  Schouten  remarks,  a 
theory  of  an  elastic  aether  in  which  at  each  point  p,  q,  ...  a  vector 
displacement  ^,  17,  ...  of  the  element  of  the  medium  is  supposed, 
involving  a  strain  and  an  elastic  stress  determined  in  terms  of 
the  strain  by  assigned  laws.  Only  it  is  to  be  remembered  that 
time  is  now  in  a  fourth  dimension,  in  which  the  historical  world- 
process  is  all  spread  out  once  for  all;  so  that  the  feature  of  elastic 
wave  propagation  becomes  a  static  relation.  The  idea  that  the 
single  fundamental  electric  vector  is  represented  by  a  superposed 
metric  is  thus  correlative  with  the  usual  dynamical  hypothesis 
that  electric  force  is  a  stress  in  an  aether.  It  thus  affords  another 
illustration  of  this  kind  of  speculation:  the  interlacing  of  space 
and  time  for  purposes  of  electrodynamics  having  upset  the  his- 
torical development  of  dynamical  principles  on  a  Newtonian  basis 
of  separate  space  and  time,  order  has  to  be  re-constituted  by 


342  Sir  Joseph  Larmor 

piecing  together  a  cognate  analytical  scheme  on  a  symmetrical 
fourfold  basis  which  tries  to  make  no  difference  between  them. 

It  is  not  improbable  that  these  remarks  merely  turn  over 
ground  that  has  already  been  explored  by  cultivators  of  hyper- 
geometry.  But  it  may  be  claimed  that  the  interest  of  this  range 
of  ideas  extends  far  beyond  the  analytical  technique,  and  that  their 
naive  expression  in  a  form  of  language  outside  its  conventions  may 
prove  to  be  helpful  in  other  regions  of  speculation. 


The  argument  above  has  been  based  on  the  supposition  that  the 
mathematical  analysis  must  establish  a  complete  correspondence, 
element  for  element,  between  the  activities  in  the  new  space-time 
and  in  the  Newtonian  space  and  time.  That  however  is  not  the  case. 
There  is  a  gravitational  correspondence  into  which  radiation  and 
its  rays  do  not  enter.  As  regards  the  latter  no  conclusions  could  be 
drawn  at  all,  except  in  the  special  circumstances  in  which  the 
coordinate  X/^  that  stands  nearest  to  time  *  does  not  enter  explicitly 
into  the  quadratic  expression  determining  the  space.  If  that  is 
postulated  the  equations  of  propagation  of  radiation  have  their 
solutions  periodic  as  regards  x^^^  treated  as  a  quasi-tim.Q,  therefore 
every  beam  of  radiation  carries  with  it  a  scale  of  X/^^  throughout 
its  course  |.  Moreover,  if  the  spacial  quadratic  contained  hx^  in  a 
product  term,  the  velocities  of  the  waves  of  radiation  in  forward 
and  backward  directions  would  not  be  the  same :  their  half  difference 
would  thus  be  the  local  velocity  of  the  frame  of  reference  in  that 
direction.  Where  hx/^^  does  not  occur  in  the  first  power,  the  frame  of 
reference  is  thus  fixed  locally  with  respect  to  the  waves  of  light 
and  their  assumed  underlying  uniform  fourfold  extension  with 
regard  to  which  they  are  propagated. 

Thus,  under  these  postulated  circumstances  of  x^  not  occurring 
explicitly  in  So-^,  the  mere  fact  that  isotropic  vibratory  radiation 
exists  with  its  absolute  velocity  c  is  sufficient,  not  merely  to  de- 
termine absolute  measurements  both  in  space  and  time,  at  every 
locality  in  the  extension,  but  also  to  determine  the  rate  of  motional 
change  of  the  coordinates  as  referred  to  the  uniform  space-time  of 
the  radiation.  It  is  gravitational  correspondence,  subject  to  this 
general  control  of  the  whole  range  of  space-time  by  observations 
of  light,  with  its  isotropic  and  uniform  qualities,  that  has  led  to 
verifiable  conclusions.  Cf.  letter  in  Nature,  Jan.  22,  1920:  also 
Monthly  Notices  R.  Astron.  Soc. 

*  That  is  the  one  coordinate  the  square  of  whose  differential  is  affected  in  dcr- 
with  a  negative  sign,  which  marks  it  off  from  the  others. 

t  It  is  the  alleged  measurement  of  this  abstract  coordinate  x^  by  a  travelling 
clock,  which  connotes  a  physical  system,  that  is  a  main  source  of  confusion. 


Gravitation  and  Light  343 

We  have  absorbed  gravitation  into  space  and  time  by  distorting 
the  latter  from  its  essential  Newtonian  uniformity:  but  there  can 
be  no  illusion  about  the  matter  either  way,  for  the  theoretical 
measuring  bar  of  the  differential  spacial  theory  is  not  our  only 
instrument;  in  the  practical  world  rays  of  light  provide  the  essential 
isotropic  measures,  and  the  spectroscope  is  always  available  to 
reveal  to  us  what  spacial  adjustments  have  been  made,  in  relation 
to  the  underlying  frame  with  regard  to  which  the  propagation  of 
light  is  isotropic  and  has  its  standard  absolute  velocity.  Light, 
instead  of  conforming  to  local  relativity,  imposes  its  own  absolute 
space-time*. 

The  argument  may  be  directed  tow^ards  yet  another  type  of 
conclusion,  as  follows.  When  change  is  made  from  Newtonian 
space  and  pure  time  to  the  uniform  space-time  fourfold,  the 
equation  of  a  straight  path  is  altered  from  h^ds  =  0  to  SJ(Zct  =  0. 
The  free  orbits  in  any  field  of  force  of  potential  energy  function 
—  F  can  readily  be  altered  so  as  to  preserve  continuity  with  this 
change,  as  above,  that  is,  so  that  where  F  becomes  negligible  they 
tend  to  straight  lines:  they  are  then  given  by 

h\{d<j^  +  2Vdt^Y  =  ^. 

The  interpretation  is  at  hand,  to  regard  them  as  the  analogues  of 
straightest  paths  in  a  modified  space-time,  referred  to  a  set  of 
coordinates  represented  now  by  colourless  symbols  x^,  x^,  Xz,  x^ 
and  given  in  terms  of  them  by 

8a2  =  Sa;i2  +  Sx^^  +  Sx^^  -  c^  (1  -  2c-2F)  Sx^^ 

As  Sct^  does  not  here  involve  x^^  explicitly,  the  differential  equations 
of  propagation  of  free  radiation,  as  expressed  in  this  space-time 
in  terms  of  these  coordinates,  have  solutions  involving  the  quasi- 
time,  x^  only  in  the  form  e'^^*:  therefore  the  radiation  from  any 
source,  however  far  it  has  travelled,  retains  the  same  period  in 
regard  to  x^  as  it  had  at  the  start.  Around  a  radiating  molecule  the 
extension  can  be  taken  as  practically  uniform:  therefore  the 
interval  of  absolute  time  is  equal  to  (1  —  c'W)  hx^.  It  follows  thus 
from  the  periodicity  as  regards  x^  that  the  periodic  time  of  a  ray 
alters  as  it  travels  so  as  to  be  proportional  to  1  —  c~^V .  If  the  ray 
belongs  to  a  definite  molecular  period  at  the  Sun,  it  has  changed 
when  it  reaches  the  Earth  so  as  to  agree  no  longer  with  that  period 
as  reproduced  by  a  local  vibrator. 

All  this  is  true  only  to  the  first  order,  but  it  applies  to  any  law  of 
potential,  and  is  irrespective  of  any  special  energy-tensor  theory. 
The  point  to  be  brought  out  is  that  if  influence  of  gravitation  on 

*  Prof.  Eddington  in  a  recent  article,  Quarterly  Review,  Jan.  1920,  seems  not  to 
disagree  with  this  conclusion:  at  any  rate  he  contemplates  the  possibility  of  an 
aether. 


344 


Sir  Joseph  LanTwr,  Gravitation  and  Light 


spectral  periods  were  definitely  disproved,  then  it  would  appear 
that  any  hope  of  bringing  orbits  into  direct  relation  with  the 
electrodynamic  space-time  fourfold  must  be  abandoned  altogether*, 
on  the  threshold.  This  drastic  conclusion  is  perhaps  an  argument  in 
fa^^our  of  the  existence  of  the  effect. 

The  other  two  verifiable  effects,  the  influence  on  the  planetary 
perihelia  and  the  deviation  of  light  passing  near  the  Sun,  arise  in 
part  from  first  order  and  in  part  from  second  order  causes.  Unlike 
the  previous  one,  their  exact  verification  is  thus  a  test  of  the  special 
theory  of  Einstein,  or  the  equivalent  Least  Action  formulation.  Its 
original  recommendation  was  that  it  restricts  the  universal  forces 
of  nature  to  the  one  type  of  gravitation:  possibly  it  would  be 
difficult  to  imagine  ways  in  which  there  could  be  room  for  any 
different  result. 


*  A  formulation  of  the  original  Nordstiom  type,  starling  from  d^Vda  =  0,  is  to 
some  degree  an  exception. 


C.  T.  R.  Wilson,  On  a  Micro-voltameter  345 


On  a  Micro-voltameter.  By  C.  T.  R.  Wilson,  M.A.,  Sidney 
Sussex  College. 

[Read  19  May  1919.] 

Experiments  were  described  with  a  mercury  voltameter,  in 
which  one  elctrode  consists  of  a  sphere  of  mercury  deposited  on 
the  end  of  a  fine  platinum  wire  and  measured  by  means  of  a 
microscope.  Quantities  of  electricity  varying  from  a  few  hundred 
electrostatic  units  to  about  one  coulomb  may  be  measured  by  it. 
The  almost  instantaneous  change  of  size  of  the  drop  when  a 
capacity  of  one  tenth  of  a  microfarad,  charged  to  1  volt,  is  dis- 
charged through  the  instrument  is  easily  observed.  A  magnet 
inserted  in  or  removed  from  a  coil  connected  to  the  terminals  of 
the  voltameter  produces  an  easily  measured  effect.  Experiments 
were  also  mentioned  which  suggest  the  possibility  of  its  application 
in  measurements  of  much  smaller  electrical  quantities. 


346    R.  Whiddington,  The  self-oscillations  of  a  Thermionic  Valve 


The  self -oscillations  of  a  Thermionic  Valve.  By  R.  Whid- 
dington, M.A.,  St  John's  College. 

[Read  19  May  1919.] 

(Abstract.) 

It  lias  been  found  possible  to  produce  oscillations  of  almost  any 
frequency  from  a  three  electrode  vacuum  valve,  without  employing 
the  usual  capacity-induction  circuits.  Thus  a.  valve  with  two 
suitable  batteries,  one  in  the  anode  circuit,  another  in  the  grid 
circuit,  will  produce  quite  powerful  oscillations,  whose  frequency 
will  be  determined  by  the  value  of  the  grid  potential. 

The  phenomenon  can  be  explained  by  supposing  that  the  oscil- 
lations are  due  to  surges  of  mercury  ions  closing  in  on  the  filament 
from  the  grid  with  a  frequency  given  by  the  approximate  formula 

2  2^         T/ 

n  = .  V 

md^        ' 

e   • 
where  —  is  the  usual  charge  to  mass  ratio,  d  is  the  radial  distance 

m  o  ^ 

filament  to  grid  and  V  is  the  positive  grid  voltage. 

Experiments  conducted  so  far  indicate  that  the  monatomic 
Hg  ion  with  one  live  charge  is  mainly  responsible. 


PROCEEDINGS  AT  THE  MEETINGS  HELD  DURING 

THE  SESSION  1918—1919. 

ANNUAL  GENERAL  MEETING. 

October  28,  1918. 
In  the  Comparative  Anatomy  Lectui'e  Room. 

Prof.  Marr,  President,  in  the  Chair. 

The  following  were  elected  Officers  for  the  ensuing  year : 

President : 

Mr  C.  T.  R.  Wilson. 

Vice-Presidents: 
Dr  Doncaster. 
Mr  W.  H.  Mills. 
Prof.  Marr. 

Treasurer: 

Prof.  Hobson. 

Secretaries : 

Mr  Alex.  Wood. 
Mr  G.  H.  Hardy. 
Mr  H.  H.  Brindley. 

Other  Members  of  Council: 
Dr  Shipley. 
Prof.  Biffen. 
Mr  L.  A.  Borradaile. 
Mr  F.  F.  Blackman. 
Prof.  Sir  J.  Larmor. 
Prof.  Eddington. 
Dr  Marshall. 
Prof.  Baker. 
Prof.  Newall. 
Dr  Fenton. 

The  following  was  elected  an  Associate  of  the  Society : 
G.  A.  Newgass,  Trinity  College. 

The  following  Communications  were  made  to  the  Society: 

1.  Proof  of  certain  identities  in  combinatory  analysis.  By  Prof.  L.  J. 
Rogers  and  S.  Ramanujan,  B.A.,  Trinity  College. 

2.  Some  properties  of  p  (n),  the  number  of  partitions  of  n.    By 
S.  Ramanujan,  B.A.,  Trinity  College. 


348  Proceedings  at  the  Meetings 

3.  On  the  exponentiation  of  well-ordered  series.  By  Miss  D.  Wrinch. 
(Communicated  by  Mr  Gr.  H.  Hardy.) 

4.  On  certain  trigonometrical  series  which  have  a  necessary  and 
sufficient  condition  for  uniform  convergence.  By  A.  E.  Jolliffe. 
(Communicated  by  Mr  Gr.  H.  Hardy.) 

5.  Some  geometrical  interpretations  of  the  concomitants  of  two 
quadrics.  By  H.  W.  Turnbull,  M.A.  (Communicated  by  Mr  G.  H. 
Hardy.) 

6.  On  Mr  Ramanujan's  congruence  properties  of  p  (n).  By  H.  B.  C. 
Darling,  B.A.    (Communicated  by  Mr  G.  H.  Hardy.) 

7.  On  the  correct  Generic  Position  of  Dacrydium  Bidwillii  Hook  f. 
By  B.  Sahni,  M.A.,  Emmanuel  College.  (Communicated  by  Professor 
Seward.) 


February  3,  1919. 
In  the  Balfour  Library. 

Mr  C.  T.  R.  Wilson,  President,  in  the  Chair. 

The  following  were  elected  Fellows  of  the  Society : 

S.  R.  U.  Savoor,  B.A.,  Trinity  College. 
S.  C.  Tripathi,  B.A.,  Emmanuel  College. 

The  following  was  elected  an  Associate : 
P.  W.  Burbidge,  Trinity  College. 

The  following  Communications  were  made  to  the  Society : 

1.  The  Gauss-Bonnet  Theorem  for  multiply-connected  regions  of  a 
surface.  By  E.  H.  Neville,  M.A.,  Trinity  College. 

2.  On  the  representations  of  a  number  as  a  sum  of  an  odd  number  of 
squares.  By  L.  J.  Mordell.   (Communicated  by  Mr  G.  H.  Hardy.) 

3.  On  certain  empirical  formulae  connected  with  Goldbach's 
Theorem.  By  N.  M.  Shah  and  B.  M.  Wilson.  (Communicated  by 
Mr  G.  H.  Hardy.) 

4.  Note  on  Messrs  Shah  and  Wilson's  paper  entitled :  On  certain 
empirical  formulae  connected  with  Goldbach's  Theorem.  By  G.  H. 
Hardy,  M.A.,  Trinity  College  and  J.  E.  Littlewood,  M.A.,  Trinity 
College. 


Proceedings  at  the  Meetings  349 

February  17,  1919. 
In  the  Comparative  Anatomy  Lecture  Room. 

Mr  C.  T.  E.  Wilson,  President,  in  the  Chair. 
The  following  Communications  were  made  to  the  Society : 

1.  Note  on  an  experiment  dealing  with  mutation  in  bacteria.    By 

Dr  DONCASTER. 

2.  Electrical  conductivity  of  bacterial  emulsions.   By  Dr  Shearer. 

3.  The  bionomics  of  Aphis  grossulariae,  Kalt.,  and  Aphis  viburni. 
Shrank.  By  Miss  M.  D.  Haviland.  (Communicated  by  Mr  H.  H. 
Brindley.) 

4.  (1)    The  conversion  of  saw-dust  into  sugar. 
(2)    Bracken  as  a  source  of  potash. 

By  J.  E.  Purvis,  M.A.,  Corpus  Christi  College. 

5.  Terrestrial  magnetic  variations  and  their  connection  with  solar 
emissions  which  are  absorbed  in  the  earth's  outer  atmosphere.  By 
S.  Chapman,  M.A.,  Trinity  College. 

6.  The  distribution  of  Electric  Force  between  two  electrodes,  one 
of  which  is  covered  with  radioactive  matter.  By  W.  J.  Harrison,  M.A., 
Clare  College. 


May  19,  1919. 
In  the  Cavendish  Laboratory. 

Mr  C.  T.  R.  Wilson,  President,  in  the  Chatr. 

The  following  were  elected  Fellows  of  the  Society : 
E.  V.  Appleton,  M.A.,  St  John's  College. 
W.  G.  Palmer,  M.A.,  St  John's  College. 
S.  P.  Prasad,  B.A.,  Trinity  College. 

The  following  was  elected  an  Associate : 

Mrs  Agnes  Arber. 
The  following  Communications  were  made  to  the  Society: 

1.  (1)    Use  of  Neon  Lamps  in  Technical  stroboscopic  work. 

(2)    The  distribution  of  intensity  along  the  positive  ray  parabolas 
of  atoms  and   molecules  of  Hydrogen  and  its  possible 
explanation. 
By  F.  W.  Aston,  M.A.,  Trinity  College. 

2.  On  a  Micro-voltameter.    By  C.  T.  R.  Wilson,  M.A.,  Sidney 

Sussex  College. 

3.  The  self-oscillations  of  a  Thermionic  Valve.  By  R.  Whiddington, 
M.A.,  St  John's  College. 


INDEX  TO  THE  PROCEEDINGS 

with  references  to  the  Transactions. 


M 


Abel's  Theorem  and  its  converses  (Kienast),  129. 

Amos,  A.,  Experimental  work  on  clover  sickness,  127. 

Aphidius,    Life    History    of   Lygocerus    (Proctotrypidae),    hyperparasite    of 

(Haviland),  293. 
Aphis  grossulariae  Kalt.,  Bionomics  of  (Haviland),  266. 
Aphis  viburni  Schr.,  Bionomics  of  (Havtland),  266. 
Appleton,  E.  v..  Elected  Fellow  1919,  May  19,  349. 
Arbek,  a.,  Elected  Associate  1919,  May  19,  349. 
Aston,  F.  W.,  Neon  Lamps  for  Stroboscopic  Work,  300. 
The  distribution  of  intensity  along  the  positive  ray  parabolas  of  atoms 

and  molecules  of  hydrogen  and  its  possible  explanation,  317. 
Axiom  in  Symbohc  Logic  (Van  Horn),  22. 

Bacteria,  Mutation  in  (Doncaster),  269. 

Bailey,  P.  G.,  see  Punnett,  R.  C. 

Bessel  functidns  of  equal  order  and  argument  (Watson),  42. 

Bessel  functions  of  large  order  (Watson),  96. 

Bionomics  of  Aphis  grossulariae  Kalt.,  and  Aphis  viburni  Schr.  (Havtland),  266. 

BORRADAILE,  L.  A.,  On  the  Functions  of  the  Mouth-Parts  of  the  Common 

Prawn,  56. 
Bracken  as  a  source  of  potash  (Purvis),  261. 
Brindley,  H.  H.,  Notes  on  certain  parasites,  food,  and  capture  by  birds  of 

the  Common  Earwig  (Forficula  auricularia),  167. 
Buchanan,  D.,  Asymptotic  Satellites  in  the  problem  of  three  bodies.    See 

Transactions,  xxii. 
Burbidge,  p.  W.,  Elected  Associate  1919,  February  3,  348. 

Cambridgeshire  Pleistocene  Deposits  (Marr),  64. 

Caporn,  a.  St  Clair,  The  Inheritance  of  Tight  and  Loose  Paleae  in  Avena  niida 

crosses,  188. 
Cells,  Action  of  electrolytes  on  the  electrical  conductivity  of  (Shearer),  263. 
Chapman,  S.,  Terrestrial  magnetic  variations  and  their  cormection  with  solar 

emissions  which  are  absorbed  in  the  earth's  outer  atmosphere.     See 

Transactions,  xxii. 
Colourimeter  Design  (Hartridge),  271. 
Convergence,  Uniform  (Jolliffe),  191. 
Convergence,  Uniform,  concept  of  (Hardy),  148. 


Index  351 

Convergence  of  certain  multiple  series  (Hardy),  86. 

Corals,  Reactions  to  Stimuli  in  (Matthai),  164. 

Crabro  cephalotes.  Solitary  wasp  (Warburton),  296. 

Cubic  Binomial  Congruences  with  Prime  Moduli  (Pocklington),  57. 

Darling,  H.  B.  C,  On  Mr  Ramanujan's  congruence  properties  of  ^  {n),  217. 

See  MacMahon,  P.  A. 

Dirichlet,  Theorem  of  (Todd  and  Norton),  111. 

DoNCASTER,  L.,  Note  on  an  experiment  dealing  with  mutation  in  bacteria,  269. 
DuTT,  C.  P.,  On  some  anatomical  characters  of  coniferous  wood  and  their 
value  in  classification,  128. 

Electric  Force  between  two  Electrodes  (Harrison),  255. 
Electrometer,  A  self-recording,  for  Atmospheric  Electricity  (Rudge),  1. 
Empii'ical  formula  comiected  with  Goldbach's  Theorem  (Shah  and  Wilson), 

238. 
Exponentiation  of  well-ordered  series  (Wkinch),  219. 

Eish-freezrng  (Gardiner  and  Nuttall),  185. 

Forficula  auriciiluria,  Common  Earwig,  parasites,  food,  and  captm-e  by  birds 
of  the  (Brindley),  167. 

Gardiner,  J.  Stanley,  and  Nuttall,  G.  H.  F.,  Fish-freezing,  185. 

Gauss-Bonnet  Theorem  (Neville),  234. 

Gibson,  C.  Stanley,  Elected  FeUow  1918,  May  20,  189. 

Goldbach's  Theorem  (Shah  and  Wilson),  238. 

Gravitation  and  Light  (Larmor),  324. 

Gray,  J.,  The  Effect  of  Ions  on  Cihary  Motion,  313. 

Green,  F.  W.,  Elected  Fellow  1916,  November  13,  126. 

Hardy,  G.  H.,  On  a  theorem  of  Mr  G.  Polya,  60. 

On  the  convergence  of  certain  multiple  series,  86. 

Sir  George  Stokes  and  the  concept  of  uniform  convergence,  148. 

See  Rogers,  L.  J.,  and  Ramanujan,  S. 

Hardy,  G.  H.,  and  Littlewood,  J.  E.,  Note  on  Messrs  Shah  and  Wilson's 

paper  entitled:  On  an  empirical  formula  connected  with  Goldbach's 

Theorem,  245. 
Hargreaves,  R.,  The  Character  of  the  Kinetic  Potential  in  Electromagnetics. 

See  Transactio7is,  xxii. 
Harrison,  W.  J.,  The  distribution  of  Electric  Force  between  two  Electrodes, 

one  of  which  is  covered  with  Radioactive  Matter,  255. 

The  pressure  in  a  viscous  liquid  moving  through  a  channel  with  diverging 

boundaries,  307.  . 
Hartridge,  H.,  Colom-imeter  Design,  271. 
Haviland,  M.  D.,  The   bionomics   of  Aphis  grossulariae  Kalt.,  and  Aphis 

viburni  Schr.,  266. 

vol.  XIX.  part  VI.  24 


352  Index 

Haviland,  M.  D.,  Preliminary  Note  on  the  Life  History  of  Lygocerus  (Procto- 

trypidae),  hyperparasite  of  Aphidius,  293, 
Hill,  M.  J.  M.,  On  the  Fifth  Book  of  Euclid's  Elements  (Foui'th  Paper).   See 

Transactions,  xxn. 
Horn,  see  Van  Horn. 

Hydrodynamics  of  Relativity  (Weatherburn),  72. 
Hydrogen  Ion  Concentration  (Saunders),  315. 

Identities  in  combinatory  analysis  (Rogers  and  Ramanitjan),  211. 

Ince,  E.  Lindsay,  Elected  Fellow  1918,  February  18,  189. 

Intensity  along  the  positive  ray  parabolas  of  atoms  and  molecules  of  hydi'ogen 

(Aston),  317. 
Ion,  Hydrogen,  Concentration  (Saunders),  315. 
Ions,  Effect  of,  on  Ciliary  Motion  (Gray),  313. 

JoLLiFFE,  A.  E.,  On  certain  Trigonometrical  Series  which  have  a  Necessary 

and  Sufficient  Condition  for  Uniform  Convergence,  191. 
Jones,  W.  Morris,  Elected  Associate  1916,  October  30,  126.  1 

KiENAST,  A.,  Extensions  of  Abel's  Theorem  and  its  converses,  129. 

Lake,  P.,  Glacial  Phenomena  near  Bangor,  North  Wales,  127. 

Shell-deposits  formed  by  the  flood  of  January,  1918,  157. 

Larmor,  J.,  Gravitation  and  Light,  324. 

Liquid,  Viscous,  pressiire  in  a  (Harrison),  307. 

LiTTLEWOOD,  J.  E.,  see  Hardy,  G.  H. 

Logic,  Primitive  Propositions  of  (Nicod),  32. 

Logic,  Symbolic,  an  axiom  in  (Van  Horn),  22. 

Lygocerus    (Proctotrypidae),    hyperparasite   of    Aphidius,    Life    History    of 

(Haviland),  293. 
Lynch,  R.  I.,  Elected  Fellow  1916,  November  13,  126. 

Exhibition  of  the  Fruit  of  Chocho  Sechium  edule,  127. 

Mackenzie,  K.  J.  J.,  see  Marshall,  F.  H.  A. 

MacMahon,  p.  a.,  On  certain  integral  equations,  188. 

MacMahon,  p.  a.,  and  Darling,  H.  B.  C,  Reciprocal  Relations  in  the  Theory 

of  Integral  Equations,  178. 
Madreporarian  Skeleton  (Matthai),  160. 
Marr,  J.  E.,  Submergence  and  glacial  cUmates  during  the  accumulation  of  the 

Cambridgeshire  Pleistocene  Deposits,  64. 
Marshall,  F.  H.  A.,  and  Mackenzie,  K.  J.  J.,  On  extra  mammary  glands  and 

the  reabsorption  of  milk  sugar,  127. 
Matthai,  G.,  Is  the  Madreporarian  Skeleton  an  Extraprotoplasmic  Secretion 

of  the  Polyps?,  160. 

On  Reactions  to  Stimuli  in  Corals,  164. 

Micro-voltameter  (Wilson),  345. 


Index  353 

Modular  Functions  (Mordell),  117. 

Moduli,  Prime,  Quadratic  and  Cubic  Binomial  Congruences  with  (Pockling- 

ton),  57. 
Mordell,  L.  J.,   On  Mr  Ramanujan's  Empirical  Expansions  of  Modular 

Functions,  117. 
Multiple  series.  Convergence  of  certain  (Hardy),  86. 
Mutation  in  bacteria  (Doncaster),  269. 

Neon  Lamps  for  Stroboscopic  Work  (Aston),  300. 

Neville,  Eric  H.,  The  Gauss-Bonnet  Theorem  for  Multiply-connected  Regions 

of  a  Surface,  234. 
Newgass,  G.  a..  Elected  Associate  1918,  October  28,  347. 
NicoD,  J.  G.  P.,  A  Reduction  in  the  number  of  the  Primitive  Propositions  of 

Logic,  32. 
Norton,  H,  T.  J.,  see  Todd,  H. 
NuTTALL,  G.  H.  F.,  see  Gardiner,  J.  Stanley. 

Oldham,  F.  W.  H.,  Elected  Fellow  1917,  February  5,  127. 

Palmer,  W.  G.,  Elected  Fellow  1919,  May  19,  349. 

Partitions  of  n,  Some  properties  of  p  {n),  the  number  of  (Ramanujan),  207, 
Phase,  Limits  of  applicability  of  the  Principle  of  Stationary  (Watson),  49. 
PoCKLiNGTON,  H.  C,  The  Du-ect  Solution  of  the  Quadratic  and  Cubic  Binomial 

Congruences  with  Prime  ModuU,  57. 
P6lya,  G.,  see  Hardy,  G.  H. 
Prasad,  S.  P.,  Elected  Fellow  1919,  May  19,  349. 
Prawn,  Common,  Functions  of  the  Mouth-Parts  (Borradaile),  56. 
Primitive  Propositions  of  Logic  (Nicod),  32. 

Proceedings  at  the  Meetings  held  during  the  Session  1916-1917,  125. 

1917-1918,  187. 
1918-1919,  347. 
PuNNETT,  R.  C,  and  Bailey,  P.  G.,  Inheritance  of  henny  plumage  in  cocks,  126. 

Some  experiments  on  the  Inheritance  of  weight  in  Rabbits,  188. 

Purvis,  J.  E.,  The  conversion  of  saw-dust  into  sugar,  259. 

Bracken  as  a  source  of  potash,  261. 

Quadratic  and  Cubic  Binomial  Congruences  with  Prime  Moduli  (Pockling- 

TON),  57. 

Quadncs,  Geometrical  Interpretations  of  the  Concomitants  of  Two  (Turn- 
bull),  196. 

Ramanujan,  S.,  Elected  Fellow  1918,  February  18,  189. 

On  the  expression  of  a  number  in  the  form  ax'^  +  hy'^  +  cz^  +  du^,  11. 

Empu'ical  Expansions  of  Modular  Functions  (Mordell),  117. 

On  certain  Trigonometrical  sums  and  their  applications  in  the  theory  of 

numbers.   See  Transactions,  xxii. 


354  Index 

Ramanujan,  S.,  On  some  definite  integrals,  188. 

Some  properties  of  ^  (n),  the  number  of  partitions  of  n,  207. 

8ee  Rogers,  L.  J. 

Reciprocal  Relations  in  the  Theory  of  Integral  Equations  (MacMahon  and 

Daeling),  178. 
Relativity,  Hydrodynamics  of  (Weatherburn),  72. 
Rodrigues,  Natural  History  of  (Snell  and  Tams),  283. 
Rogers,  L.  J.,  and  Ramanujan,  S.,  Proof  of  certain  identities  in  combinatory 

analysis,  211. 
RuDGE,  W.  A.  D.,  A  seK-recording  electrometer  for  Atmospheric  Electricity,  1. 


the  I 


Sahni,  B.,  On  an  Australian  specimen  of  Clepsy  drop  sis,  128. 

Observations  on  the  Evolution  of  Branching  in  the  Ferns,  128. 

■ ■  On  the  branching  of  the  Zygopteridean  Leaf,  and  its  relation  to 

probable  Pinna-nature  of  Oyropteris  sinuosa,  Goeppert,  186. 

•  The  Structure  of  Tmesipteris  Vieillardi  Dang,  186. 

— — •  On  Acmopyle,  a  Monotypic  New  Caledonian  Podocarp,  186. 
Saunders,  J.  T.,  On  the  growth  of  Daphne,  126. 

A  Note  on  Photosynthesis  and  Hydrogen  Ion  Concentration,  315. 

Savoor,  S.  R.  U.,  Elected  Fellow  1919,  February  3,  348. 

Saw-dust,  Conversion  of,  into  sugar  (Purvis),  259. 

Shah,  N.  M.,  and  Wilson,  B.  M.,  On  an  empirical  formula  connected  with 

Goldbach's  Theorem,  238. 
Shearer,  C,  The  action  of  electrolytes  on  the  electrical  conductivity  of  the 

bacterial  cell  and  their  effect  on  the  rate  of  migration  of  these  cells  in 

an  electric  field,  263. 
Shell-deposits  formed  by  the  flood  of  January,  1918  (Lake),  157. 
Snell,  H.  J.,  and  Tams,  W.  H.  T.,  The  Natiu-al  History  of  the  Island  of 

Rodrigues,  283. 
Stokes,   Sir   George,    and   the   concept   of   uniform   convergence   (Hardy), 

148. 
Symbolic  Logic,  An  Axiom  in  (Van  Horn),  22. 


Tams,  W.  H.  T.,  see  Snell,  H.  J. 

Theorem  of  Dirichlet  (Todd  and  Norton),. 111. 

Theorem  of  Mr  G.  Polya  (Hardy),  60. 

Thermionic  Valve,  Self-oscillations  of  a  (Whiddington),  346. 

Todd,  H.,  and  Norton,  H.  T.  J.,  A  particular  case  of  a  theorem  of  Dirichlet, 

111. 
Trigonometrical  Series  which  have  a  Necessary  and  Sufficient  Condition  for 

Uniform  Convergence  (Jolliffe),  191. 
Tripathi,  S.  C,  Elected  Fellow  1919,  February  3,  348. 
TuRNBULL,  H.  W.,  Some  Geometrical  Interpretations  of  the  Concomitants  oi 

Two  Quadrics,  196. 

Van  Horn,  C.  E.,  An  Axiom  in  Symbohc  Logic,  22. 


i 

hlet,  I 


I 


Index  355 

Waeburton,  C,  Note  on  the  solitary  wasp,  Crabro  cephaloies,  296. 

Wasp,  Crabro  cephalotes  (Warburton),  296. 

Watson,  G.  N.,  Bessel  functions  of  equal  order  and  argument,  42. 

The  hmits  of  applicability  of  the  Principle  of  Stationary  Phase,  49. 

Bessel  functions  of  large  order,  96. 

Asymptotic  expansions  of  hypergeometric  functions.    See  Transactions, 

xxn. 
Weatherburn,  C.  E.,  On  the  Hydrodynamics  of  Relativity,  72. 
WmDDiNGTON,  R.,  The  self-oscillations  of  a  Thermionic  Valve,  346. 
Wilson,  B.  M.,  see  Shah,  N.  M. 
Wilson,  C.  T.  R.,  Methods  of  investigation  in  atmospheric  electricity,  126. 

On  a  Micro-voltameter,  345. 

Wood,  T.  B.,  The  siu-face  law  of  heat  loss  in  animals,  126. 
Woods,  H.,  The  Cretaceous  Faunas  of  New  Zealand,  127. 
Wrinch,  D.,  On  the  exponentiation  of  well-ordered  series,  219. 

Yamaga,  N,  Elected  Associate  1916,  November  13,  126. 


CAMBRIDGE:    PRINTED  BY  .1.   B.  PEACE,  M.A.,  AT  THE  UNIVERSITY  PRESS 


I 


CONTENTS. 

PAGE 

Colourimeter  Design.     By  H.   Hartridge,   M.D.,   Fellow   of  King's 

College,  Cambridge.     (One  Fig.  in  text) 271 

The  Natural  History  of  the  Island  of  Rodrigues.  By  H.  J.  Snell 
(Eastern  Telegraph  Company)  and  W.  H.  T.  Tams.  (Communi- 
cated by  Professor  Stanley  Gardiner) 283 

Preliminary  Note  on  the  Life  History  of  Lygocerus  {Proctotrypidae), 
hyperparasite  of  Aphidius.  By  Maud  D.  Haviland,  Fellow  of 
Newnbam  College.     (Communicated  by  Mr  H.  H.  Brindley)       .       293 

Note  on  the  solitary  wasp,  Crabro  cephalotes.     By  Cecil  Warburton, 

M.A.,  Christ's  College 296 

Neon  Lamps  for  Stroboscopic  Work.  By  F.  W.  Aston,  M.A.,  Trinity 
College  (D.Sc,  Birmingham),  Clerk-Maxwell  Student  of  the  Uni- 
versity of  Cambridge.     (One  Fig.  in  text) 300 

The  pressure  in  a  viscous  liquid  moving  through  a  channel  loith  diverging 
boundaries.  By  W.  J.  Harrison,  M.A.,  Fellow  of  Clare  College, 
Cambridge.     (One  Fig.  in  text) 307 

The  Efect  of  Ions  on  Ciliary  Motion.     By  J.  Gray,  M.A.,  Fellow  of 

King's  College,  Cambridge 313 

A  Note  on  Photosynthesis  and  Hydrogen  Ion  Concentration.     By  J.  T,  ■ 

Saunders,  M.A.,  Christ's  College 315 

The  distribution  of  intensity  along  the  positive  ray  parabolas  of  atoms 
and  molecules  of  hydrogen  and  its  possible  explanation.  By  F.  W. 
Aston,  M.A.,  Trinity  College  (D.Sc,  Birmingham),  Clerk-Maxwell 
Student  of  the  University  of  Cambridge.     (Three  Figs,  in  text)    .       317 

Gravitation  and  Light.     By  Sir  Joseph  Larmor,  St  John's  College, 

Lucasian  Professor 324 

On  a  Micro-voltameter.     By  C.  T.  R.  Wilson,  M.A.,  Sidney  Sussex 

College 345 

The  self-oseillations  of  a  Thermionic  Valve.    By  R.  Whiddington,  M,  A., 

St  John's  College 346 

Proceedings  at  the  Meetings  held  duiing  the  Session  1918 — 1919         .       347 

Index  to  the  Proceedings  with  references  to  the  Transactions      .         .       350 


PEOCEEDINGS 

OF   THE 

CAMBKIDGE  PHILOSOPHICAL  SOCIETY 

VOLUME   XX 


PRINTED  IN  ENGLAND 

AT  THE  CAMBRIDGE  UNIVERSITY  PRESS 

BY    J.    B.    PEACE,    M.A. 


PKOCEEDINGS 

OF  THE 

CAMBEIDGE  PHILOSOPHICAL 
SOCIETY 


VOLUME  XX 

26  January  1920—16  May  1921 


CAMBRIDGE 

AT  THE  UNIVERSITY  PRESS 

and  sold  by 

deighton,  bell  &  co.,  ltd.  and  bowes  &  bowes,  cambeidge 

cambridge  university  press 
c.  f.  clay,  manager,  fetter  lane,  london,  e.c.  4 

1921 


I 


CONTENTS. 

VOL.  XX. 


PAGE 


On  the  term  by  term  integration  of  an  infinite  series  over  an  infinite  range 
and  the  inversion  of  the  order  of  integration  in  repeated  infinite 
integrals.  By  S.  Pollard,  M.A.,  Trinity  College,  Cambridge. 
(Communicated  by  Prof.  6.  H.  Hardy) 1 

Note  on  Mr  Hardy^s  extension  of  a  theorem  of  Mr  Polya.    By  Edmuxd 

Landau.   (Communicated  by  Prof.  G.  H.  Hardy)    ....       14 

Studies  on  Cellulose  Acetate.    By  H.  J.  H.  Fenton  and  A.  J.  Berry       .       16 

An  examination  of  Searle's  method  for  determining  the  viscosity  of  very 
viscous  liquids.  By  Kurt  Molin,  Filosofie  Licentiat,  Physical 
Institute,  Technical  College,  Trondhjem.  (Communicated  by  Dr 
G   F.  C.  Searle.)    (Four  figs  in  Text) 23 

Preliminary  Note  on  Antennal  Variation  in  an  Aphis  (Myzus  ribis, 
Linn.).  By  Maud  D.  Haviland,  Fellow  of  Newnham  College. 
(Communicated  by  Mr  H.  H.  Brindley)         .....       35 

The  effect  of  a  magnetic  field  on  the  Intensity  of  spectrum  lines.  By  H.  P. 
Waran,  M.A.,  Government  Scholar  of  the  University  of  Madras. 
(Communicated  by  Professor  Sir  Ernest  Rutherford.)  (Plates  I 
and  II  and  one  fig.  in  Text)       ........       45 

Further  Notes  on  the  Food  Plants  of  tite   Common  Earwig  (Forficula 

auricularia).    By  H.  H.  Brindley,  M.A.,  St  John's  College      .        .       50 

Lagrangian  Methods  for  High  Speed  Motion.    By  C.  G.  Darwin      .         .       56 

A  hifilar  method  of  measuring  the  rigidity  of  wires.  By  G.  F.  C.  Searle, 
Se.D.,  F.R.S.,  University  Lecturer  in  Experimental  Physics.  (Five 
figs,  in  Text) 61 

The  Rotation  of  the  Non-Spinning  Gyrostat.    By  G.  T.  Bennett,  M.A., 

F.R.S.,  Emmanuel  College,  Cambridge      ......       70 

Proof  of  the  equivalence  of  different  mean  vahies.    By  Alfred  Kienast. 

(Communicated  by  Professor  G.  H.  Hardy)     .....       74 

Notes  on  the  Theory  of  Vibrations.  (1)  Vibrations  of  Finite  Amplitude. 
(2)  A  Theorem  due  to  Routh.  By  W.  J.  Harrison,  M.A.,  Fellow  of 
Clare  College ■ 83 

Experiments  with  a  plane  diffraction  grating.  By  G.  F.  C.  Searle, 
Sc.D.,  F.R.S.,  University  Lecturer  in  Experimental  Physics.  (Ten 
figs,  in  Text) 88 

The  Shadow  Electroscope.    By  R.  Whiddington,  M.A.,  St  John's  College. 

(One  fig.  in  Text) 109 

Mathematical  Notes.   By  Professor  H.  F.  Baker  and  C.  V.  Hanumanta  : 
On  the  Hart  circle  of  a  spherical  triangle       .         .         .         .         .         .116 

On  a  property  of  focal  conies  and  of  bi circular  quartics         .         .         .     122 
On  the  construction  of  the  ninth  point  of  intersection  of  two  plane  cubic 
curves  of  which  eight  points  are  given  ......     131 


vi  Contents 


Mathematical  Notes  (continued); 

On  a  proof  of  the  theorem  of  a  double  six  of  lines  by  projection  from 
four  dimensions.    (Three  figs,  in  Text)       .         .         .         .         .         .133 

145 

147 


On  transformations  urith  an  absolute  quadric  .... 

On  a  set  of  transformations  of  rectangular  axes.    (One  fig.  in  Text) 

On   the  generation  of  sets  of  four  tetrahedra   of  which  any  tivo  are 

mutually  inscribed      ..........  155 

On  the  reduction  of  homography  to  movement  in  three  dimensions.    (One 

fig.  in  Text) 158 

On   the   transformation   of  the   equations   of   electrodynamics   in   the 
Maxwell  and  in  the  Einstein  for ms     .         .         .         .         .         .         .166 

On  the  stability  of  periodic  motions  in  general  dynamics        .         .         .  181 

On  the  stability  of  7'otating  liquid  ellipsoids  ......  190 

On  the  general  theory  of  the  stability  of  rotating  masses  of  liquid  .         .  198 

Sur  le  principe  de  Phragmen-Lindelijf    Par  Marcel  Riesz.    With  Note 

by  G.  H.  Hardy 205 

A    note  on   the   nature  of  the  carriers  of  the  Anode  Rays.    By   G.   P. 

Thomson,  M.A.,  Fellow  of  Corpus  Christi  College    ....     210 

Proceedings  at  the  Meetings  held  during  the  Session  1919 — 1920   .         .     212 

The  Problem  of  Soaring  Flight.  By  E.  H.  Hankin,  M.A.,  Sc.D.,  late 
Fellow  of  St  John's  College,  Cambridge,  Chemical  Examiner  to 
Government,  Agra,  India.  (Communicated  by  Mr  H.  H.  Brindley.) 
With  an  Introduction  by  F.  Handlby  Page,  C.B.E.,  F.R.Aer.S.     .     219 

Preliminary  Note  on  the  Superior   Vena  Ca,va  of  the  Cat.    By  W.  F. 

Lanchester,  M.  a.,  King's  College,  and  A.  G.  Thacker  .         .         .     228 

A  Note  on  Vital  Staining.    By  F.  A.  Potts,  M.A.,  Trinity  Hall.    (One 

fig.  in  Text) 231 

Preliminary  Note  on  a  Cynipid  hyperparasite  of  Aphides.  By  Maud  D. 
Haviland,  Fellow  of  Newnham  College.  (Communicated  by 
Mr  H.  H.  Brindley) 235 

A  method  of  testing  Triode  Vacuum  Tubes.    By  E.  V.  Appleton,  M.A., 

St  John's  College.    (Two  figs,  in  Text) 239 

The  Rotation  of  the  Non-Spinning  Gyrostat.   By  Sir  George  Greenhill 

and  Dr  G.  T.  Bennett 243 

On  the  representation  of  the  simple  group  of  order  660  as  a  groiop  of  linear 
substitutions  on  5  symbols.  By  Dr  W.  Burnside,  Honorary  Fellow 
of  Pembroke  College  .........     247 

On  the  representation  of  algebraic  numbers  as  a  sum  of  four  squares.    By 

L.  J.  Mordell.    (Communicated  by  Professor  H.  F.  Baker)  .         .     250 

On  a  Gaussian  Series  of  Six  Elements.  By  L.  J.  Rogers.   (Communicated 

by  Professor  G.  H.  Hardy) 257 

Note  on  Ramanujan^s  trigonometrical  function  Cg(n),  and  certain  series  of 

arithmetical  functions.   By  Professor  G.  H.  Hardy  ....     263 

On  the  distribution  of  primes.  By  H.  Cramer,  Stockholm.  (Communicated 

by  Professor  G.  H.  Hardy) 272 

Note  on  the  parity  of  the  number  which  emimerates  the  partitions  of  a 

number.   By  Major  P.  A.  MacMahon '     .     281 

Note  on  constant  volume  explosion  experiments.    By  S.  Lees,  M.A.,  St 

John's  College.    (Two  figs,  in  Text) 285 


Contents  vii 

PAGE 

On  the  Latent  Heats  of   Vaporisation.    By  Eric   Keightley  Rideal, 

M.A.,  Trinity  Hall ....     291 

Oil  the  fimction  [x].  By  ViGGO  Brun  (Drobak,  Norway).  (Communicated 

by  Professor  G.  H.  Hardy) 299 

A  theorem  concerning  summahle  series.    By  Professor  G.  H.  Hardy  .         .     304 

Standing  Waves  parallel  to  a  Plane  Beach.    By  H.  C.  Pocklington,  M.A., 

St  John's  College 308 

The  Origin  of  the  Disturbances  in  the  Initial  Motion  of  a  Shell.    By  R.  H. 

Fowler  and  C.  N.  H.  Lock.   (One  fig.  in  Text)        .        .        .         .311 

Tides  in  the  Bristol  Channel.   By  G.  I.  Taylor,  F.R.S.   (Four  figs,  in  Text)     320 

Expenments  ivith  Rotating  Fluids.    By  G.  I.  Taylor,  F.R.S.  .         .         .     326 

Experiments  on  focal  lines  formed  hy  a  zone  plate.  By  G.  F.  C.  Searle, 
So.D.,  F.R.S.,  University  Lecturer  in  Experimental  Physics.  (Five 
figs,  in  Text) 330 

The  Tensor  Form  of  the  Equations  of  Viscous  Motion.  By  E.  A.  Milne, 

B.A.,  Trinity  College 344 

Insect  Oases.    By  C.  G.  Lamb,  M.A 347 

A  Note  on  the  Hydrogen  Ion  Concentration  of  some  Natural  Waters.   By 

J.  T.  Saunders,  M.A.,  Christ's  College 350 

The  Mechanism  of  Ciliary  Movement.   By  J.  Gray,  M.A.,  Balfour  Student, 

and  Fellow  of  King's  College,  Cambridge.    (Three  figs,  in  Text)      .     352 

A   Note  on  the  Biology  of  the  '■  Crown-GaW  Fungus  of  Lucerne.    By 

J.  Line,  M.A.,  Emmanuel  College.    (Seven  figs,  in  Text) .         .         .     360 

On  some  Alcyonaria  in  the  Cambridge  Museum.  By  Sydney  J.  Hickson, 
M.A.,  F.R.S.,  Professor  of  Zoology  in  the  University  of  Manchester. 
(One  fig.  in  Text) 366 

The  Influence  of  Function  on  the   Conformation  of  Bones.    By  A.   B. 

Appleton,  M.A.,  Downing  College.    (Five  figs,  in  Text)  .         .         .     374 

Animal  Oecology  in  Deserts.    By  P.  A.  Buxton,  M.A.,  Fellow  of  Trinity 

College,  Cambridge 388 

Venational  Abnormalities  in  the  Diptera.  By  C.  G.  Lamb,  M.A.  (Four- 
teen figs,  in  Text) 393 

The   Cooling  of  a  Solid  Sphere  with  a  Concentric  Core  of  a  Different 

Material.    By  Professor  H.  S.  Carslaw.    (Three  figs,  in  Text)     .     399 

Symbolical  Methods  in  the  theory  of  Conduction  of  Heat.    By  Dr  T.  J.  I'a. 

Bromwich,  F.R.S.    (Two  figs."  in  Text) 411 

0?!.  the  effect  of  a  magnetic  field  on  the  intensity  of  spectrum  lines.  By 
H.  P.  Waran,  M.A.,  Government  of  India  Scholar  of  the  University 
of  Madras.  (Communicated  by  Professor  Sir  Ernest  Rutherford, 
F.R.S.)    (Three  figs,  in  Text  and  Plate  III) 428 

On  a  property  of  focal  conies  and  of  bicircular  quartics.  Qj  C.  V.  Hand- 
MANTA  Rao,  LTniversity  Professor,  Lahore.  (Communicated  by 
Professor  H.  F.  Baker) 434 

Convex  Solids  in  Higher  Space.    By  Dr  W.  Burnside,  Honorary  Fellow 

of  Pembroke  College  .........     437 

Note  on  the    Velocity  of  X-ray  Electrons.    By  R.  Whiddington,  M.A. 

(One  fig.  in  Text) 442 

A    Laboratory    Valve  method  for  determining   the    Specific    Indxictive 

Capacities  of  Liquids.  By  R.  Whiddington,  M.A.  (One  fig.  in  Text)     445 


viii  Contents 


The  Theoretical  Value  of  Stitherland' s  Constant  in  the  Ki-netic  Theory  of^ 
Gases.  By  C.  G.  F.  James,  Trinity  College,  Cambridge.  (Communi- 
cated by  Mr  R.  H.  Fowler.)    (One  fig.  in  Text)       .         .         .         .447 

On  the  Stability  of  the  Steady  Motion  of  viscous  liquid  contained  between 
two  rotating  coaxal  circidar  cylinders.  By  W.  J.  Harrison,  M.A., 
Fellow  of  Clare  College,  Cambridge 455 

The  soaring  flight  of  dragon-flies.    By  E.  H.  Hankin,  M.A.,  Sc.D.,  Agra, 

India.    (Three  figs,  in  Text) 460 

The    Gluteal    Region    of    Tarsius    Spectrum.     By    A.    B.    Appleton.  11! 

(Plate  IV) 4^^     11 

An  unusual  type  of  mcde  secondary  characters  in  the  Diptera.    By  C.  G. 

Lamb,  M.A.   (Four  figs,  in  Text) 475 

A  Note  on  the  Mouth-parts  of  certain  Decapod  Crustaceans.  By  L.  A. 
BoRRADAiLE,  M.A.,  Fellow  and  Tutor  of  Selwyn  College,  Cambridge, 
and  Lecturer  in  Zoology  in  the  University       .....     478 

An  Apparatus  for  Projecting  Spectra.    By  H.  Hartridge       .         .         .     480 

Note  on  true  and  apparent  hermaphroditism  in  sea-urchins.   B}'  J.  Gray, 

M.A.,  Balfour  Student,  Cambridge  University 481 

On  Certain  Simply -Transitive  Permutation-Groups.  By  Dr  W.  Burnside, 

Honorary  Fellow  of  Pembroke  College 482 

Proceedings  at  the  Meetings  held  during  the  Session  1920 — 1921    .         .     485 

Index  to  the  Proceedings  with  references  to  the  Transactions         .         .     492 


PLATES. 

Plates  I— III.    To  illustrate  Mr  Waran's  papers     ....       48,  433 
Plate  IV.    To  illustrate  Mr  Appleton's  paper 474 


CORRECTION. 

MoRDELL,  p.  250,  line  5,  after  conjugate  numbers  insert  in  the  reed 
conjugate  fields. 


PEOCEEDINGS 

OF  THE 

CAMBRIDGE  PHILOSOPHICAL 
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PROCEEDINGS 

OF   THE 

Cambriirgc  ISljtlasapIjkal  Bomi^. 


On  the  term  by  term  integration  of  an  infinite  series  over  an 
infinite  range  and  the  inversion  of  the  order  of  integration  in 
repeated  infinite  integrals.  By  S.  Pollard,  M.A.,  Trinity  College, 
Cambridge.    (Communicated  by  Prof.  G.  H.  Hardy.) 

[Received  1  January,  1920.     Read  8  March,  1920.] 

The  problem  for  infinite  series. 

1.  The  problem  to  be  solved  is  that  of  determining  conditions, 
under  which  the  equation 

00  /•=»  /"»     00 

2        Un(oc)dx==l     S  Un(oi;)dx,  (1) 

n=l J  a  J an=l 

is  true.  It  is  discussed  in  detail  in  Bromwich's  Infinite  Series, 
pp.  452-455,  where  various  conditions  are  given.  All  these  con- 
ditions will  be  found  to  involve  uniform  convergence,  the  fact 
being  that  the  infinite  integrals  there  considered  are  obtained  as 
limits  of  Riemann  integrals  and,  in  the  theory  of  the  latter,  con- 
siderations as  to  the  validity  of  the  equation 
rb  m  rb  oo 

lim    I     2    M„  (x)  dx  =  j    S  Uji  (w)  dx,  (2) 

almost  always  involve  uniform  convergence.  Thus  conditions  for 
term  by  term  integration  over  an  infinite  range,  being  built  up 
from  the  conditions  for  term  by  term  integration  over  a  finite 
range,  involve  uniform  convergence. 

Now  the  condition  of  uniform  convergence  is  by  no  means  a 
necessary  one :  it  occurs  because  of  the  lack  of  power  in  the 
methods  of  the  Riemann  theory.  Much  wider  conditions  can  be 
obtained  by  the  use  of  the  Lebesgue  theory.  It  is  the  object  of 
this  paper  to  give  these. 

VOL.   XX.   PART  L  1 


2  Mr  Pollard,  On  the  term  by  term  integration 

Conditions  for  passage  to  the  limit  under  the  sign  of 
integration,  the  range  of  integration  being  finite. 

2.  We  give,  for  the  sake  of  reference,  the  two  principal 
elementary  conditions. 

(C  1)  If  u^ (sc)  is  positive  for  a^x  ^b;  n  =  l,  2,  3  . . .,  then  if 
either  side  of  (2)  is  finite  the  equation  holds,  and  if  either  side  is 
infinite  both  are. 

V 

(C  2)     Ifl'^Un  (oc)  I  <  i|r  (a;)  for  a-^a:^b,v=l,2,S,...,  tuhere 
n=l 

'>^  is  summable  in  (a,  b),  then  both  sides  of  (2)  eccist  and  are  finite 
and  equal*. 

Resume  of  theorems  of  double  limits. 

3.  As  the  use  of  double  limits  is  fundamental  in  the  theory 
about  to  be  developed,  we  give  a  short  summary  of  the  results 
required. 

(a)    If  the  double  limit     lim     S^,  y  exists,  and  lim  8x,  y  exists 

for  all  sufiiciently  large  y ;  then  lim  (lim  S^,  y)  exists  and  is  equal 

to  the  double  limit.    Similarly  for  the  limit  lim   (lim  S^^y). 

(^)    If  Sx,  y  is  increasing  in  x  and  y,  and  any  one  of 
lim     Sx,y,     lim  (lim  *S^a;,  j/X     lim  (lim>Sa;,  j,) 

exist;  then  all  three  exist  and  are  equal. 

(y)  If  8x,  y  can  be  expressed  as  the  difference  of  two  functions 
S'x,  y,  S"x,  y  ^^ch  of  which  is  increasing  in  x  and  y  and 

lim        {S'xy  +  S"xy) 
X-»-oo ,  y^"Xi 

exists  and  is  finite ;  then 

lim      Sx,y,     lim  (lim  Sx,y\     lim  (lim  Sa;,y) 

all  exist  and  are  finite  and  equal. 

The  condition  (7)  is  especially  convenient  when 

S.,y=n''fil,v)d^dv. 

J  aJb 

*  De  la  Valine  Poussin,  Cours  d^analyse  infinites imale,  t.  i. ,  3rd  Ed.,  p.  264, 
theorems  iii  and  11. 


of  an  infinite  series  over  an  infinite  range  3 

For  if  lim         f"  ^  \f{l  v)\d^dr) 

exists  and  is  finite,  then  S^^y  satisfies  the  condition  of  (7).     We 
have  in  fact 

^x,  y  —  ^  x,y        ^   x,yj 
fx  ry 
where  S'x,y=       I    \f{^,v)\d^dr}, 

J  aJb 

S\y=r  ("[\f{^,V)\-f{lv)]d^dv, 
J  aJ  b 

and  both  S'cc,y,  8"x,y  are  increasing  in  x  and  y  and  have  a  finite 
double  limit — the  former  by  hypothesis  and  the  latter  because 

Note.    The  above  results  still  hold  when  either  or  both  of  the 
variables  x,  y  take  only  positive  integral  values. 

Definition  of  infinite  integrals. 

4.    Let /(a;)  be  any  function  which  is  summable  in  (a,  X)  for 
all  X  greater  than  a. 

If  lim       f{x)  dx, 

X-*-ix  J  a 

where  the  integral  is  taken  in  the  sense  of  Lebesgue,  exists  and 
is  finite,  we  say  that 


J  0 


f  (x)  dx 


converges  and  attribute  to  it  the  value  of  the  limit. 

This  definition  is  evidently  consistent  with  and  more  general 
than  that  usually  given,  where /(a:;)  is  assumed  to  be  integrable  in 
Riemann's  sense  in  (a,  X).  It  has  the  special  advantage  of  not 
being  restricted  to  functions  which  are  bounded  in  every  (a,  X). 
And  we  lose  nothing  by  adopting  it,  as  the  two  theorems  on  which 
the  theory  of  infinite  integrals  rests,  the  first  and  second  mean 
value  theorems,  are  still  true  when  we  abandon  the  restriction 
that  f{x)  is  to  have  a  Riemann  integral  and  make  only  the 
assumption  that /(a;)  is  summable*. 

General  theorems. 

rX  m 


rX  m 
5.    I.    If  the  double  limit      lim         I     2  m„  (x)  dx  exists  and 

m-^-oo,  X-*-<x>J  an=l 

(a)     I    Un  (x)  dx  converges  for  all  n, 
J  a 


Ibid.  t.  II.  2nd  Ed.,  p.  53. 


4  Mr  Pollard,  On  the  term  by  term  integration 

00 

(b)  2  u^  (x)  converges  for  X'^a, 

n=\ 

rX  m  rX  CO 

(c)  lim         S  Un  (oc)  dx,  2  ?/„  {x)  dx, 

m-*-ao  J  a  n=l  J  an=l 

exist  and  are  equal  for  all  X ;  then  both  sides  of  {!)  exist  and  are 
equal. 

rX  m 

Proof.     Write  S  u^  {cc)  dx  =  Sm,  x, 

J  a  n=l 

and  let  lim      Sm,x  =  S. 

m-*-'x>,X^'OD 


Since  I    w„  («)  dx  converges  for  all  n 

J  a 

rX  m 
lim         %  u^{x)dx,  i.e.  lim  Sm,. 


X 

exists  for  all  m.    Hence 


lim  (lim  >S^,x)  ='S^  (3)» 

Wl^-oo   X-^cc 

by(«)-  . 

In  virtue  of  (b)  and  (c) 

rX  m  ^ 

lim  1     2  Un  (^)  dx 

m^-as  J  a  n=l 
rX  00 
exists  and  is  equal  to        2  u^  (x)  dx. 
Jan=l 

Thus  lira  Sm,  x  exists  for  X  ^  a. 

Hence  lim  (lim  Sm,x)  exists  and  is  equal  to  S. 

X-^-CC     77l-*-00 

rX  00 
Taking  lim  S^,  x  in  the  form        2  u^  {^)  dx,  we  see  that 

m-*-oo  J  an  =  l 

/-■»   00 

S  Un  (^)  dx  =  S.  (4) 

J  an=l 

And  (3)  and  (4)  give  us  our  theorem. 

00 

II.    J/  2  I  w„  (x)  I  converges  for  x^a  and  the  double  limit 

rXm 


rXm 

lim  2  I  w„  (a;)  I  dx 

■00,  X->-<x)  J  an=\ 


exists  and  is  finite;  then  without  further  condition  both  sides  of  (1) 
exist  and  are  finite  and  equal. 


o/  an  infinite  series  over  an  infinite  range  5 

rx  m 
Proof.   By  (7),  if       lim       I     S  |  w„  {x)  \  dx  exists,  so  does 

«-»■», X-*-ao  J  a  n=X 
rX  m 

lim  S  w„  {x)  dx. 

m-^-oo , X-*-<x>  J an=l 
CX  m 

Also  Sm, x=\     ^  \Un{x)\dx^ 8, 

J  a  n  =  l 

for  all  X  and  m.    But  Sm,x  increases  with  X  for  each  m.    Hence 

rx  m 

%  \u„(x)\ 
J  an=l 


rX  m 

lim    I     2  I  w„  (ic)  I  dx 
X^oo  J an=l 

exists  for  each  m  and  is  less  than  8. 
And  therefore 

rX  f   rx  m  rX  m-1  "j 

lim   1     I  u„  (x)  I  dx  =  lim  j  I     S  1 1*„  (x)  \dx  —  I       %    \Un{x)\dx>  , 
X^-co  J  a  X^i-co  [J  an=l  J  a    n=l  J 

exists  for  each  m,  i.e.  1    |  ^^^  (/»)  |  dx  and  therefore  I    ?/.„  (x)  dx  con- 

^  a  J  a 

verges  for  each  m.    This  is  (a)  of  (I). 

Again,  8m,  x  increases  with  m  for  each  X. 

Hence  lim  S.m,,x  exists  and  is  finite  for  each  X.    So  from 

(C  1)  [X  00 

X    \Un (x)  I  dx 
Jan=l 

00 

is  finite.    Thus  2  |  m„  (x)  I  is  summable  in  (a,  X).    But 

mm  00 

71=1  n  =  l  71  =  1 

and  so  by  (C  2) 

rx  m  rx  CO 

lim   I     S    Un  (x)  dx,  I     2  w„  (a;)  c?^, 

m-*-oo  .' a  n.  =  l  ./a7i.=  l 

exist  and  are  finite  and  equal.    This  is  (c)  of  (I).    Now  (b)  of  (I) 
is  satisfied  by  hypothesis. 

Thus  all  the  conditions  of  (I)  are  satisfied  and  so  both  sides  of 
(1)  exist  and  are  equal. 

Deductions  from  the  general  theorems. 
6.    A.   //  u^{x)  =  (f>(x)f„{x), 

where  2  /„  (x)  converges  for  x'^a, 

n  =  l 

V 

I  ^  fn{x)\<  G,  for  x'^a  and  all  v, 

n  =  l 


6  Mr  Pollard,  On  the  term  by  term  integration 


and       I  ^  {x)  I  dx  converges;  then  both  sides  0/  (1)  exist  and  are 

J  a 

finite  and  equal. 
B.    If  either  of 

%    i    \un(cc)\dx,  2  I  M„  (x)  I  dx, 

w  =  l  J  a  J  a  n  =  l 


eadst  and  are  finite;  then  both  sides  of  (1)  exist  and  are  finite  and 
equal. 


rX  m  rx  <x> 

C    If  lim         S  Un  (x)  dx,  'S.  Un  (^)  dx, 

m-»-oo  .' a«,=  l  ■!an=l 

exist  and  are  finite  and  equal,  and 

2      Un  (x)  dx 

J  a 
converges  uniformly  for  a^^  x,  and  each 

u^  (x)  dx 

J  a 

converges;  then  both  sides  of(l)  exist  and  are  finite  and  equal. 

V 

D.    7/  I  2  M„  («)  I  <  -^x  (^)  for  a^  x^  X  and  all  v,  where  -^x 
n=l 

is  summable  in  {a,  X),  and 

rx 

S         Un (x)  dx 

J  a 

converges  uniformly  for  a^  X,  X  being  arbitrary,  and  each 

Un  {x)  dx 
J  a 

converges;  then  both  sides  of  {1)  exist  and  are  finite  and  equal. 

D  is  a  special  case  of  C  obtained  by  making  use  of  (C  2). 
A,  B,  D  may  be  regarded  as  generalisations  of  theorems  A — C, 
pp.  452-455  of  Bromwich's  Infinite  Series. 

Proofs.     A.    If  m'  >  m, 

m'  m'  m 

we  have  2    /„  {x)  =  X  fn{x)-X  fn  {x), 

and  therefore 

m'  Til'  m 

I      2     fn{x)\^\   Xfn{x)\  +  \    ^  f,{x)\^2G. 


of  an  infinite  series  over  an  infinite  range  7 

Hence 

rX'      m'  rX'        m' 

I  S        Un  {x)dx\^\      I     S     </>  {x)f^  (x)  I  dx 

JX   n=m+l  J  X    n=m+l 

rX'  m' 

<  \<^{x)\t    \fn  {OC)  I  dx 

J  X  n  =  m 

^2g[     \d>  {x)  I  dx. 

J  X 

Now,  given  any  positive  number  e,  we  can,  since        \<f)  {x)\dx 

J  a 
converges,  find  Xo  such  that 

\(f){x)\dx<  € 

J  X 

forX,  X'>Xo.    Hence 

rX'      to' 
I  1  S      ii„  (x)  dx\<  € 

Jx   n  =  m+l 

for  X,  X'  >  Xf)  and  all  rn,  m .    Thus  the  double  limit 


exists.    Further 


rx  m 
lim         /      S  i<„  ix)  dx 
•■»,X-*"X  J  a  n  =  \ 


n=l  x=l 

and   G^  1  0  (ic)  I  is   summable  in  C«,  X)  for  all  X  greater  than  a. 
Thus  by  (C  2) 

CX    m  rX    00 

lim         X  Un  (x)  dx,  i   u„  (x)  dx 

J  a  n=l  J  a  n=l 

exist  and  are  equal  and  finite. 

All  the  conditions  of  (I)  are  now  satisfied,  and  our  theorem 
follows. 

B.    If  we  write  r^  m 


rX   m 
Sm,X=  I         2    \Un{x)\dx, 
•I  a  n=l 


then      lim      8m,x  exists  and  is  either  finite  or  positive  infinity. 

In  the  first  case  our  theorem  follows  at  once  by  (II). 
In  the  second  case,  both  the  repeated  limits 

lim  (lim  Sm,x),     lim  (lim  8m,  x), 

m-^cD  X^'x>  X^oD  m-*oo 

are  infinite.     Suppose  now  that 

S    I    I  M„ (x) I dx 

»=1  J  a 


8  Mr  Pollard,  On  the  term  by  term  integration 

exists  and  is  finite.     Then    lim  (lim  ^«t,x)  exists  and  is  finite, 
and  we  get  a  contradiction.    And  if 

^    \Un{x)\dx 
J  an=l 

exists  and  is  finite,  then  so  does 

rx  m 
\      2   I  ti„  {x)  I  dx 

J  a  7i=l 

for  all  X  greater  than  a.     Hence  as  in  theorem  II 
rX   oa  rX  TO 

I       "E  \  Un  {w)  \  dx  =  lim  X  I  Wn  (^)  I  ^^> 

J  a  71=1  Wi-»-oo   J  a  71  =  1 

and  it  follows  that 

r""     00 

E  \un\x)\dx=  lim  (lim  S^^x), 

Ja  n  =  l  X-*-oo  7j,-*-oo 

and  we  again  get  a  contradiction. 

Thus  the  first  case  alone  is  possible,  and  this  is  the  case  in 
which  our  theorem  is  true. 

C.    Write  r  u,{w)dx  =  g,{X), 

-  « 

m 
tgn{X)^Sm,X. 

71=1 

00 

Since  S  g^  (X)  converges  uniformly  for  a  ^  X,  given  e  >  0  we 

71=1 

can  find  No  such  that 

I     i    gAX)\<^,        {X>a,N^N-,). 

n=N+l 

Thus         \Sm,x-  i  9n  (X)  I  <  e,    {X^a,  m  ^  N^). 

71  =  1 

00 

Hence  if  lim    S  gn{^)  exists  and  is  finite, so  does      lim      aS'^^^ x , 

X-*-^  n=l  7?l-»Q0  ,  JC-».Xi 

and  the  two  are  equal.    Now 

I  i:  g^  (ZO  -i  g,  (Z")  I  ^  i  f  g,  (Z)  -  %  g,  (Z")  j 

tt=l  71=1  71=1  n=l 

+  1       i       gn{X')\  +  \       i       5'n(X")I 
n=N+l  n=N+l 

^1  f  5r,(Z')-i5'«(X")l  +  26. 

71=1  71=1 

But  since  lim  g^  (Z)  (=       u^  (w)  dx)   exists  and  is  finite  for 

X-*oo  J  a 


o/  an  infinite  series  over  an  infinite  range  9 

N 

each  n,  so  does  lim    S  gn{^)  ^-nd  we  can  find  X^  such  that 
1  f  5r„  (Z')  -I  g,  {X")  1  <  6.  {X',  X"  ^  Zo). 

71=1  »=1 

Hence     1  I  ^„  (X)  -  2  ^„  (Z")  |  <  3e,       (Z',  Z"  ^  Z„) 
and  therefore,  by  the  general  principle  of  convergence 

00 

lim    2  gr,  {X) 

X-»-t»    71=1 

exists.  Thus  lim  *S^„i,x  exists.  The  other  conditions  of  (I)  are 
satisfied  by  hypothesis  and  our  theorem  follows. 

The  problem  for  infinite  integrals. 
7.     We  have  to  determine  conditions  under  which  the  equation 

dx\    f{x,y)dy==\    dy  \   f{x,y)dx  (5) 

J  a         J  b  J  b         J  a 

is  true.  The  methods  adopted  above  apply  almost  without  change 
and  we  get  conditions  almost  identical  with  those  already  given. 
We  quote  them  without  "proof,  as  the  proofs  can  be  made  up 
immediately  on  the  lines  of  those  already  given. 

As  regards  the  nature  oif{x,  y),  we  assume  throughout  that 
f  {x,  y)  is  summable  in  the  region 

{a^x^X,  b^y^Y), 
for  all  Z  ^  a,  Y  ^b  ;  so  that,  by  Fubini's  theorem*,  the  repeated 

rX  rY  fY         rX 

integrals         dx  \    f{x,y)dy,     I     dy  \   f{x,y)dx   exist   and   are 

J  a  ■  b  J  b  J  a 

equal  to  the  double  integral. 

General  theorems. 

rX   rY 


and 


8.     I'.    If  the  double  limit     lim        I     j    f{x,  y)  dxdy  exists 

(a)  I    f{x,y)dx,         converges  for  y^b,         * 

(b)  I     f{x,y)dy,         converges  for  x '^  a, 

rX        rY  rX         r^ 

(c)  lim       dx      f{x,  y)  dy,         dx      f{x,  y)  dy, 

F-w-oo  J  a         J  b  J  a         J  b 

*  De  la  Valine  Poussin,  Integrales  de  Lebesgue  etc.,  p.  53. 


10  Mr  Pollard,  On  the  term  by  term  integration 

exist  and  are  finite  and  equal  for  X  ^  a ;  then  both  sides  of  (5) 
exist  and  are  equal. 

IT.    If  the   double   limit       lim  \  f{x,  y)\dxdy  exists 

X-*-ao ,  Y-^x  J  a    J  b 

and  is  finite;  then  without  further  condition  both  sides  of  (5)  exist  and 
are  finite  and  equal. 

Deductions  from  the  general  theorems. 
9.     A'.    //  f{x,y)  =  <^{x)d{x,y), 

where  j       6  {x,  y)  dy  \  <  G  for  x^a,  y  ^b, 

Jh 

f{x,  y)  dy  converges  for  x^a,' 

'b 

and       \(f>{x)\dx  converges;  then  both  sides  of  (5)  exist  and  are 

J  a 

finite  and  equal. 
B'.     If  either  of 

1    dx      \f{x,  y)  I  dy,        dy       \f{x,  y)  \  dx, 

J  a  J  b  J  b        J  a 

exist  and  are  finite;  then  both  sides  of  {5)  exist  and  are  finite  and 
equal*. 

rX        rY  rx        r=°  ■ 

C.    If  lim        dxl   f{x,y)dy,  dx  \  f{x,y)dy, 

Y-*-oo  J  a         J  b  J  a  J  b 

exist  and  are  finite  and  equal,  and 

dy      fix,  y)  dx 

Jb         Ja 

converges  uniformly  for  a^  X,  and 

o 

f{x,  y)  dx 


/ 

J  a 


converges  for  y  ^  b ;  then  both  sides  of  (o)  exist  and  are  finite  and 
equal. 

*  This  is  de  la  Valine  Poussin's  theorem.    See  Bromwieh,  Infinite  Series,  p.  457. 
The  hypothesis  given  by  Bromwieh  to  the  effect  that  both  the  integrals 


/"OO  /"OO 


are  convergent  is  unnecessary,  the  existence  of  one  (the  one  necessary  to  the 
existence  of  the  repeated  integral)  is  sufficient.  That  of  the  other  is  implied  by 
the  existence  of  the  double  hmit,  see  Note  2. 


o/  an  infinite  series  over  an  infinite  range  1 1 

D'.    If  \r fix,  y)dy\^^x{x)  for  a^  x^  X,  6  ^  F, 

J  b 

ivhere  yjrx  is  summahle  in  (a,  X),  and 

\    dy  \   f  {x,  y)  dx 
Jb         J  a 

converges  uniforndy  for  a  ^  X,  X  being  arbitrary,  and 

f{x,  y)  dx 


converges  for  y^b;  then  both  sides  of  {b)  exist  and  are  finite  and 
equal. 

10.  Note  1.  Results  B  are  especially  valuable,  as  they  are 
easy  to  remember  and  convenient  to  apply.  The  power  of  the 
Lebesgue  theory  is  shewn  very  clearly  here  in  that  by  using  it 
we  are  enabled  to  make  the  hypothesis  which  ensures  the  exist- 
ence of  the  double  limit*  ensure  also  the  passage  to  the  limit  under 
the  sign. 

Note  2.  It  is  well  to  be  precise  as  to  the  meaning  of  the  word 
"exists"  as  used  in  connection  with  repeated  Lebesgue  integrals. 

Suppose /(^,  y)  is  measurable  in  x,  y  in  the  rectangle 

^a^x^X\ 
Kb^y^Yj' 

We  know  that  the  function  y (a:;,  y)  considered  as  a  function  of 
X,  is  measurable  in  (a,  X)  for  each  y  in  (6,  Y)  a  set  of  zero  measure 
being  excepted.     It  may  not,  however,  be  summable  in  (a,  X),  i.e. 

f{x,y)dx 


may  not  exist,  for  all  ?/ concerned.    But,  if/(^,  y)  is  summable 
over  the  rectangle,  i.e.,  if  the  double  integral 

rx  rY 

fix,  y)dxdy 

■  b 

exists ;  then  it  can  be  shewn  that 

rx 
fix,  y)dx 


f 

J  a 


exists  for  all  values  of  y  in  (b,  Y),  save  possibly  those  of  a  set  of 
measure  zero. 

CO      ,-co 

*  The  existence  of    2    I      j  u^^  (x)  \  dx  implies  the  existence  of  the  double  limit 
n=lj  a 

by  (7)  of  §3  ;  and  in  addition,  by  the  use  of  (C  1)  on  |  u„(x)  |  ,  it  will  be  found  to 
imply  the  validity  of  the  passage  to  the  limit  under  the  sign. 


12  Mr  Pollard,  On  the  term  by  term  integration 

Now  in  the  Lebesgue  theory  the  integral  of  any  summable 
function  over  a  set  of  zero  measure  is  zero,  and  consequently  we 
may  neglect  a  set  of  measure  zero  without  affecting  the  value  of  _ 
the  integral.  Hence  when  we  are  faced  with  the  problem  of  find-  l{ 
ing  the  value  of  a  function  which  is  indefinite  or  infinite  at  the 
points  of  a  set  of  measure  zero,  we  simply  neglect  these  points 
and  find  the  value  of  the  integral  over  the  residue.  This  is  taken 
to  be  the  value  of  the  integral  over  the  original  set. 

With  the  above  convention  it  is  true  that,  if 

I         /(^,  y)  dxdy 
Ja  Jb 

exists,  so  does  dy      f{x,  y)  dxdy, 

.  h  J  a 

although  there    may  be   points   in   (6,    Y)  at  which  the  single 

integral 

rx 
f{x,y)dx 


f 

J  a 


does  not  exist. 

It  is  always  to  be  understood  in  dealing  with  repeated 
Lebesgue  integrals  (finite  or  infinite)  that  the  inner  integrals 
need  only  exist  at  all  the  points  of  the  range  of  integration  of  the 
outer  integral  save  those  of  a  set  of  measure  zero. 

Let  us  apply  the  foregoing  remarks  to  theorem  B'.    Suppose 

rY        rx 

dy  \  \f{^,y)\dx 

■lb         J  a 

exists.    Then  we  know  that 

lim         11  fix,  y)  I  dxdy 

X-*-x ,  F-9.20  J  a  J  b 

exists.    It  follows  that 


I    dx\    \f{x,y)\dy, 

J  a         Jb 

on  of  Y,  is  bounded  £ 
lim   /    dx      \f{x,y)\dy 

T-^x  J  a         J  b 


considered  as  a  function  of  Y,  is  bounded  as  F  tends  to  infinity. 
Thus 

rx        rY 


Y- 
exists  and  is  finite.    It  follows  that 


\f{x,  y)  \  dy 

b 

converges  at  all  the  points  of  (a,  X)  save  possibly  those  of  a  set  of 


of  an  infinite  series  over  an  infinite  range  13 

measure   zero,   because  if  it  did  not  the  above  limit  would  be 
infinite  ;  and  so,  for  our  purposes 


J  a         J  b 


exists. 

Our  convention  has  enabled  us  to  infer  the  existence  of  the 
inner  integrals  from  the  existence  of  the  double  limit. 

Note  3.  A  thorough  treatment  on  different  lines  of  the  subject 
of  this  paper  will  be  found  in  two  papers  by  Prof  W.  H.  Young : 

(1)  "  On  the  change  of  order  of  integration  in  an  improper 
repeated  integral,"  Trans.  Gamh.  Phil.  Soc,  xxi.  p.  361. 

(2)  "The  application  of  expansions  to  definite  integrals," 
Proc.  Lond.  Math.  Soc,  ix.  (1910),  p.  463. 

In  this  paper  we  content  ourselves  with  giving  simple 
generalisations  of  well-known  results  with  proofs  depending  on 
comparatively  elementary  theorems.  There  is  no  attempt  to 
obtain  comprehensive  results. 


14     Mr  Landau,  Note  on  Mr  Hardy's  extension  of  a  theorem,  etc. 


Note  on  Mr  Hardy's  extension  of  a  theorem  of  Mr  Poly  a. 
By  Edmund  Landau.    (Communicated  by  Prof.  G.  H.  Hardy.) 

[Received  10  December  1919.    Read  26  January  1920.] 

In  a  recent  note  in  these  Proceedings*  Mr  Hardy  has  estab- 
lished an  improved  form  of  a  theorem  of  Mr  Pdlya,  viz. : 

Suppose  that  g  {x)  is  an  integral  function,  and  M  (r)  the  maxi- 
mum of  I  ^  (^)  I  for  \x\^r.  Suppose  further  that  g  {x)  is  an  integer 
for  x  =  0,  1,  2,  3,  ...,  and  that 

M{r)=o{2'). 
Then  g  (x)  is  a  polynomial. 

As  Mr  Hardy  remarks  at  the  beginning  of  his  note,  it  is 
sufficient  (after  the  analysis  given  already  by  Mr  Pdlya),  to  prove 
the  one  formula 

^,2-r  _ — ^ — =0(1). 


—  TT 


n  (2n  -  s  cos  6) 

Mr  Hardy's  proof  of  this  formula  may  be  replaced  by  the  following 
shorter  proof 
Since 

nl2^n^ =  nl2^n(^^-^,2^nn^l=OWn), 

n(2n-s)  ^  ^' 

s=l 

it  is  enough  to  prove 

'[ir{e,n)de==o{^), 

where 

ylr(e,n)=u(^  ^^  ~  '    . 
^  ^       '     s=i\2n  —  scos6 

Now 

l-cos6/_2-24  +  ---^2      24^2        24  ~  12 ' 
for  —  TT  <  ^  ^  TT,  and 


1  +  2/ 

*  Vol.  XIX.  (1919),  pp.  60-63. 


Mr  Landau,  Note  on  Mr  Hardy's  extension  of  a  theorem,  etc.     15 
for  0  ^  y  ^  1.    Hence 

1-7?  1  1 


1-^cos^      l+_^(l-cos^)      ^+va-cos0) 

1  -7] 

^  g-i^  (1-cosfl)   <  g-Tll*^ 

for  0  ^  t;  ^  |,  —  TT  ^  6  ^tt;  and  so 

1  —  — -  02      n 

-^{d,  n)  =  n  — —  ^  e~487tif  =  e-TyV(«+i)9^  ^  e-?Vne^^ 

*='l-^cos6' 
In 

for  —  TT  ^  ^  ^  TT  and  ??  =  1,  2,  3,  ....    Therefore 

["  A/r  (^,  n)de^j      e-  ^^^''dd  =  0  ('-^)  . 

GoTTlNGEJf,  4  December  1919. 


16     Dr  Fenton  and  Mr  Berry,  Studies  on  Cellulose  Acetate 


Studies   on    Cellulose   Acetate.     By    H.    J.    H.    Fenton    aii( 
A.  J.  Berry. 

[Read  8  March  1920.] 

The  enormous  demand  for  cellulose  acetate  and  the  serious, 
shortage  of  acetone  and  certain  other  materials  used  in  the  manu- 
facture of  aeroplane  dopes  during  the  war  originated  a  systematic 
research  on  cellulose  acetate,  especially  as  regards  the  behaviour 
of  this  material  towards  solvents  and  its  chemical  properties 
generally.  The  research  has  been  pursued  in  a  number  of  directions, 
the  most  important  of  which  have  been  (a)  substitutes  for  acetone 
as  solvents,  (b)  the  preparation  of  cellulose  acetate  and  a  study  of 
the  influence  of  the  mode  of  preparation  on  the  properties  of  the 
resulting  product,  and  (c)  the  analytical  chemistry  of  cellulose  ace- 
tate. Most  of  our  experiments,  especially  those  relating  to  aeroplane 
dopes  were  necessarily  of  a  technical  character,  but  as  a  few  results- 
of  general  chemical  interest  have  been  obtained  in  the  course  of 
the  work,  we  have  thought  it  desirable  to  give  a  brief  account  of 
them  in  the  present  communication. 

Solvents. 

At  the  time  of  the  difficulty  caused  by  the  serious  shortage  of 
acetone  we  were  urged  to  discover  efficient  substitutes  for  this, 
solvent  for  use  in  aeroplane  dopes.  It  should,  in  passing,  be 
observed  that  the  properties  of  acetone  make  it  an  ideal  solvent: 
its  conveniently  low  boiling  point,  rapid  solvent  action  on  cellulose 
acetate,  non-poisonous  character,  and,  in  normal  times,  cheap 
and  abundant  supply.  All  other  liquids  which  have  so  far  been 
suggested  show  a  deficiency  in.  some  one  or  other  of  these 
particulars. 

In  August,  1917,  we  suggested  that  in  case  of  emergency  the 
three  following  solvents  might  be  employed,  viz.  aeetaldehyde^ 
acetonitrile,  and  nitrobenzene  with  certain  additions.  Quite  early 
in  the  investigation  (October,  1916)  we  suggested  acetic  acid  and 
ethyl  formate  as  solvents.  We  also  suggested  the  use  of  cyclo- 
hexanone  and  of  beechwood  creosote  as  substitutes  for  tetrachloro- 
ethane  or  benzyl  alcohol  as  high  boiling  solvents.  We  were  never 
informed  whether  these  solvents  were  actually  employed.  It  is- 
remarkable  that  at  considerably  later  dates,  patents  have  been 
taken  out  for  the  use  of  both  acetaldehyde  and  cyclohexanone  as- 
dope  constituents.  (British  Patent  131647,  July  4th,  1918  (acet- 
aldehyde) and  Ibid.  130402,  February  15th,  1918  (Cyclohexanone).) 


Dr  Fento7i  and  Mr  Berry,  Studies  on  Cellulose  Acetate      17 

Our  experiments  have  demonstrated  that  the  destructive  effect 
of  acids  upon  fabrics  is  dependent  on  the  strength  of  the  acid  in 
the  physico-chemical  sense.  Hitherto  it  had  been  supposed  that 
esters  were  objectionable  as  dope  constituents  on  account  of  the 
possibilities  of  free  acids  resulting  from  hydrolysis.  This,  however, 
we  found  not  to  be  the  case.  As  far  as  weak  acids  only  are  concerned, 
tensile  strength  determinations  gave  excellent  results;  and  fabrics 
doped  with  acetic  acid  as  the  principal  solvent  compared  most 
favourably  with  others. 

In  our  experiments  a  large  number  of  liquids  have  been 
examined,  not  only  from  the  purely  practical  point  of  view,  but 
also  from  a  desire  to  obtain  if  possible  some  information  with 
regard  to  possible  relationships  between  the  nature  of  the  liquid 
and  its  solvent  action.  It  is  of  course  impossible  to  define  strictly 
the  solubility  of  cellulose  acetate  in  any  given  solvent  owing  to  the 
colloidal  nature  of  the  products.  The  term  "positive"  is  used  in 
the  following  lists  to  imply  that  the  liquid  named  has  the  property 
of  gelatinizing  cellulose  acetate  and  subsequently  converting  it 
into  a  clear  homogeneous  "sol"  without  the  aid  of  heat.  All  the 
results  were  obtained  with  a  sample  of  the  material  which  yields 
54  per  cent,  of  acetic  acid  on  cold  alkaline  saponification. 

Positive. 

Liquid  ammonia,  liquid  sulphur  dioxide,  liquid  hydrogen 
cyanide,  acetaldehyde,  benzaldehyde,  salicylaldehyde,  acetone, 
methyl  ethyl  ketone,  suberone,  acetonitrile,  propionitrile,  formic 
acid,  acetic  acid,  butyric  acid,  formamide,  ethyl  formate,  ethyl 
oxalate,  ethyl  malonate,  etbyl  acetoacetate,  aniline,  phenyl- 
hydrazine,  ortho-toluidine,  piperidine,  pyridine,  tetrachloroethane, 
nitrobenzene*,  nitromethane,  cyclohexanone,  guaiacol,  chloro- 
form*. 

Although  cellulose  acetate  is  insoluble  in  water  and  in  absolute 
ethyl  alcohol,  a  mixture  of  these  two  liquids  dissolves  it  freely 
on  boiling.  On  cooling,  however,  precipitation  takes  place  almost 
completely. 

Negative. 

Liquid  air,  liquid  ethylene,  liquid  nitrous  oxide,  liquid  hydrogen 
sulphide,  benzene,  toluene,  turpentine,  carbon  disulphide,  carbon 
tetrachloride,  alcohol,  ether,  ethyl  chloride,  acetal,  dimethyl 
acetal,  nickel  carbonyl,  and  many  other  liquids. 

No  general  conclusion  can  be  drawn  as  regards  the  chemical 
nature  of  a  liquid  and  its  solvent  action  on  cellulose  acetate.   It  is, 

*  Nitrobenzene  requires  certain  additions.  Chloroform  had  only  a  partial 
solvent  action  on  this  specimen  of  the  material. 

VOL.  XX.  PART  I.  2 


18     Dr  Fenton  and  Mr  Berry,  Studies  on  Cellulose  Acetate 

however,  worthy  of  note  that  there  appears  to  be  some  relation 
(with  undoubted  exceptions)  between  the  dielectric  constant  and 
solvent  action. 

Influence  of  methods  of  preparation  upon  the  properties 
of  cellulose  acetate. 

The  materials  obtained  by  acetylating  cellulose  with  acetic 
anhydride  diluted  with  acetic  acid  in  presence  of  various  catalysts 
such  as  concentrated  sulphuric  acid,  ferric  sulphate,  ortho  tolui- 
dine  bisulphate,  may  show  considerable  variations  in  properties 
depending  upon  the  temperature,  length  of  time  of  acetylation, 
and  numerous  other  factors.  When  cellulose  is  acetylated  and 
the  product  at  once  precipitated  by  water,  it  is  nearly  insoluble  in 
acetone.  Various  methods  have  been  adopted  in  order  to  convert 
the  product  so  obtained  into  an  acetone-soluble  modification.  The 
most  widely  used  of  these  methods  is  that  of  Miles.  This  consists  in 
heating  the  acetic  acid  solution  of  the  cellulose  acetate  with  water 
in  rather  greater  quantity  than  that  required  to  combine  with  the 
residual  acetic  anhydride.  Sodium  acetate  may  also  be  added  to 
react  with  the  catalyst  if  still  present.  The  results  are  usually 
supposed  to  be  due  to  chemical  hydration. 

In  our  experiments,  cellulose  was  acetylated  under  the  influence 
of  various  catalysts,  and  the  effect  of  treatment,  by  the  Miles 
process  was  subjected  to  a  critical  examination.  The  most  marked 
effects  of  this  process  are  the  changes  in  solubility  in  acetone  and 
chloroform,  most  cellulose  acetates  being  soluble  in  chloroform 
and  insoluble  in  acetone  before  the  treatment.  This  change  in 
physico-chemical  properties  was  found  to  be  accompanied  by  a 
fall  in  the  acetyl  number.  In  one  case  the  untreated  cellulose 
acetate  with  an  acetyl  number  of  60-9,  yielded  a  product  after  the 
Miles  process  carried  out  at  100°  for  48  hours  with  an  acetyl 
number  of  46-7.  In  another  case  when  the  treatment  was  carried 
out  at  the  same  temperature  for  23  hours,  the  acetyl  number  fell 
from  60-5  to  50-4.  The  specific  gravity  of  the  cellulose  acetate  is 
also  greatly  reduced  after  the  treatment.  The  influence  on  the 
heat  test  is  not  well  marked  but  the  decomposition  point  appears 
to  be  lowered  somewhat. 

In  our  view  these  results  are  to  be  ascribed  to  partial  hydrolysis 
of  the  cellulose  esters,  not  to  hydration  as  is  commonly  supposed*. 
Apart  from  the  diminution  of  the  acetyl  number  already  mentioned, 
we  have  carried  out  a  series  of  experiments  which  have  demon- 
strated that  cellulose  acetate  does  not  form  a  hydrate.    These 

*  Oux  view  that  the  effect  of  the  Miles  process  is  essentially  hydrolytic  and  not 
due  to  chemical  hydration  has  been  expressed  subsequently  by  Ost  {Zeifsch. 
angeic.  Chem.  1919,  xxxii,  66,  76,  and  82). 


Dr  Fenton  and  Mr  Berry,  Studies  on  Cellulose  Acetate      19 

experiments  originated  in  connexion  with  our  determinations  of 
tlie  water  contained  in  commercial  samples  of  cellulose  acetate. 
As  is  well  known,  the  water  is  readily  expelled  by  exposure  of  the 
material  over  concentrated  sulphuric  acid  in  a  desiccator  or  by 
heating  to  100°.  It  has  frequently  been  supposed  that  the  approxi- 
mately constant  proportion  of  5  or  6  per  cent,  of  water  usually 
met  with  indicates  a  definite  hydrate.  In  order  to  obtain  positive 
information  on  this  point,  we  determined  the  pressure-concentra- 
tion relationship  in  the  manner  originally  adopted  by  van  Bemmelen 
in  his  well  known  researches  on  silicic  acid  {Zeitsch.  anorg.  Chem. 
1896,  XIII.  233).  Weighed  quantities  of  the  material  were  exposed 
in  a  series  of  exhausted  desiccators  over  sulphuric  acid  of  various 
determined  concentrations,  and  the  corresponding  vapour  pressures 
were  found  by  reference  to  Landolt  and  Bornstein's  tables.  The 
weights  were  found  to  be  constant  after  24-48  hours,  and  the 
pressure  concentration  relationship  showed  that  no  chemical 
hydration  occurs.  The  phenomenon  is  to  be  regarded  as  one  of 
adsorption,  probably  with  subsequent  difiusion,  and  is  precisely 
similar  to  the  absorption  of  water  by  cellulose  itself.  (Compare 
Masson  and  Richards  (Proc.  Roy.  Soc.  1906,  lxxviii.  421),  Trouton 
and  Pool  {Ibid.  1906,  lxxvii.  292)  and  Travers  {Ibid.  1906,  lxxviii. 
21,  and  1907,  lxxix.  204).) 

Characterization  and  Analysis  of  cellulose  acetate. 

In  the  technical  analysis  of  cellulose  acetate,  it  is  usual  to 
examine  the  product  by  the  heat  test,  solubility,  acidity,  and 
viscosity  of  the  solutions,  in  addition  to  the  determinations  of 
acetyl  (as  acetic  acid),  copper  reducing  power,  water,  ash,  and 
impurities.  We  have  made  an  exhaustive  investigation  of  various 
methods  of  carrying  out  these  determinations,  especially  of  the 
acetyl  number,  and  have  also  carried  out  many  ultimate  analyses 
for  carbon  and  hydrogen  in  some  commercial  specimens  of  the 
material. 

The  methods  of  determining  the  acetyl  group  may  be  classified 
under  the  two  heads  of  alkaline  saponification  and  acid  hydrolysis. 
In  the  former  the  substance  is  saponified  by  excess  of  standard 
alkali,  either  at  the  ordinary  temperature  or  at  some  higher  tem- 
perature, and  the  excess  of  alkali  determined  by  titration.  In  the 
latter,  the  substance  is  hydrolysed  by  strong  acid,  usually  sulphuric 
or  phosphoric,  and  the  resulting  acetic  acid  separated  by  steam 
distillation  (Ost),  or  alcohol  is  added  and  the  resulting  ethyl  acetate 
distilled  off  and  collected  in  excess  of  standard  alkali  (Green  and 
Perkin).  The  following  is  a  summary  of  the  principal  results 
obtained  in  our  experiments. 

(1)  Cold  alkaline  saponification  (Ost  and  Katayama,  Zeitsch. 
angew.  Chem.  1912  (25),  1467).    A  known  weight  of  the  substance 

2 2 


20     Dr  Fenton  and  Mr  Berry,  Studies  on  Cellulose  Acetate 

is  soaked  with  alcohol,  then  a  measured  volume  of  normal  alkali 
is  added  and  allowed  to  stand  for  24  hours.  The  excess  of  alkali 
is  then  determined  by  standard  acid.  The  mean  result  was  54  per 
cent,  of  acetic  acid  calculated  for  the  dry  substance. 

(2)  Cold  alkaline  saponification  (Boeseken,  van  der  Berg  and 
Kerstjens,  Rec.  Trav.  Chim.  1916,  xxxv.  320).  The  substance  is 
treated  with  strong  aqueous  potash  for  one  or  two  days.  A  measured 
excess  of  normal  hydrochloric  acid  is  then  added,  the  liquid  then 
boiled  for  a  moment  to  expel  carbon  dioxide  and  the  resulting 
solution  titrated  with  baryta  water.  The  mean  result  calculated  as 
above  was  53-5  per  cent,  of  acetic  acid. 

(3)  Hot  alkaline  saponification  (Barthelemy,  Moniteur  Scienti- 
fique,  1913  (3),  ii.  549).  In  this  method  the  saponification  is  effected 
by  heating  the  substance  with  normal  soda  for  about  16  hours  at 
85°.  The  excess  of  alkali  is  then  determined  by  titration  with 
standard  acid.  Several  experiments  were  made  in  which  the  condi- 
tions were  subjected  to  considerable  variations  as  regards  length 
of  heating  and  amount  of  excess  of  alkali.  The  extreme  variations 
in  the  acetyl  number  calculated  as  above  were  60-0  and  62-1  per 
cent. 

(4)  Hot  alkaline  saponification  (Green  and  Perkin,  Trans. 
Chem.  Soc.  1906,  812).  The  saponification  is  carried  out  at  the 
boiling  point  with  semi-normal  alcoholic  soda  and  the  excess  of 
alkali  titrated  by  standard  acid.  Our  experiments  yielded  results 
of  60  per  cent,  of  acetic  acid,  the  extreme  variations  being  58-2 
and  61-9  per  cent.  These  numbers  are  in  agreement  with  those  of 
Green  and  Perkin  (loc.  cit.). 

It  is  evident  that  the  methods  of  hot  alkaline  saponification 
invariably  yield  results  which  are  considerably  higher  than  those 
obtained  by  cold  saponification.  There  can  be  little  doubt  that 
the  higher  results  are  due  to  the  action  of  alkali  on  the  regenerated 
cellulose.  Support  to  this  contention  was  obtained  by  digesting  two 
equal  weights  of  filter  paper  with  50  c.c.  of  normal  soda  for  two 
days,  one  at  the  ordinary  temperature,  the  other  at  85°.  In  the 
former  case  no  alkali  was  consumed,  while  the  heated  product 
showed  a  loss  of  nearly  2  c.c.  of  normal  alkali  on  titration. 

(5)  Acid  hydrolysis  (Ost,  loc.  cit.).  The  substance  is  first 
digested  with  50  per  cent,  (by  volume)  sulphuric  acid.  After  24 
hours  the  liquid  is  diluted  considerably  and  the  acetic  acid  separated 
by  steam  distillation,  and  titrated  with  baryta  water.  In  our 
experiments  phosphoric  acid  was  substituted  for  sulphuric  acid  in 
order  to  avoid  error  due  to  possible  formation  of  sulphur  dioxide. 
The  results  varied  from  51-5  to  55-0  per  cent,  of  acetic  acid. 

(6)  Acid  hydrolysis  (A.  G.  Perkin,  Trans.  Chem.  Soc.  1905,  107). 
In  this  method  the  cellulose  acetate  is  treated  with  ethyl  alcohol 
and  sulphuric  acid,  and  the  resulting  ethyl  acetate  distilled  into 


Dr  Fenton  and  Mr  Berry,  Studies  on  Cellulose  Acetate      21 

excess  of  standard  alkali.  The  ester  is  then  saponified  and  the  excess 
of  alkali  determined  by  titration.  In  our  experiments  phosphoric 
acid  was  used  instead  of  sulphuric  acid  for  the  reason  already 
mentioned.  The  results  varied  from  52-2  to  54-4  per  cent,  of  acetic 
acid. 

In  our  opinion,  preference  should  be  given  to  the  method  of 
cold  alkaline  saponification  of  Ost.  Not  only  are  the  results  more 
uniform,  but  they  agree  well  with  those  obtained  by  acid  hydro- 
lysis. The  latter  methods  are  exceedingly  tedious  to  carry  out. 
We  have  also  carried  out  some  experiments  with  the  use  of  hot 
baryta  water  as  a  saponifying  agent  and  subsequent  gravimetric 
determination  of  the  barium,  the  results  averaging  57-58  per  cent, 
of  acetic  acid. 

The  materials  met  with  in  commerce  known  as  cellulose  acetate 
are  most  probably  mixtures  or  solid  solutions  of  various  acetates, 
not  definite  chemical  individuals.  If,  however,  it  were  desired  to 
represent  cellulose  acetate  as  a  chemical  individual,  the  results  of 
our  analyses  of  a  number  of  specimens  do  not  correspond  with  the 
formula  of  the  triacetate  C6H7O2  (0C0CH3)3  which  is  commonly 
supposed.  They  agree  better  with  the  formula  of  a  pentacetyl 
derivative  of  C12H20O10  and  still  better  with  that  of  a  heptacetyl 
compound  of  CjgHgQOig. 

Thus 

Carbon  Hydrogen  Acetic  acid 

CcH-Oa  (OCOCH3)3  requires  50-0  5-5  62-1  per  cent. 

C12H15O5  (OCOCH3)5    „  49-4  5-6  560 

C.sHaA  (OCOCHs)^     „  49-2  5-64  53-8 

Our  most  reliable  results  average  carbon  49-2,  hydrogen  5-5,  and  acetic 
acid  54  per  cent. 

Certain  authors  have  stated  that  sodium  ethylate  may  be  used 
for  the  determination  of  acetyl  in  cellulose  acetates.  In  investi- 
gating this  reaction,  we  were  surprised  to  find  that  ethyl  acetate 
was  always  produced  along  with  a  yellow  sodium  derivative  of 
cellulose.  Quantitative  experiments  were  performed  in  which  the 
ethyl  acetate  was  distilled  into  an  excess  of  standard  sodium 
hydroxide,  and  after  saponification  determined  with  standard 
acid.  The  residue  was  washed  with  alcohol  to  remove  the  unaltered 
sodium  ethylate  and  this  solution  was  titrated  with  standard 
acid.  The  residue  was  then  treated  with  water  to  decompose  the 
sodium  compound  and  titrated  also.  It  was  found  that  the  quantity 
of  acetic  acid  converted  into  ethyl  acetate  to  that  becoming  sodium 
acetate  appears  to  depend  to  some  extent  on  the  proportion  of 
sodium  ethylate  employed.  The  results  can  be  explained,  if  the 
average  commercial  cellulose  acetates  are  represented  by  the 
formula  C12H15O5  (OCOCH3)5,  by  the  equation: 


22     Dr  Fenton  and  Mr  Berry,  Studies  on  Cellulose  Acetate 

C12H15O5  (0C0CH3)5  +  CgHsONa  +  4C2H5OH 

=  CiaHigOgONa  +  5CH3COOC2H5 

which  may  be  taken  to  represent  the  main  reaction. 

In  support  of  this,  the  yellow  sodium  compound  from  a  similar 
experiment,  after  thorough  washing  with  alcohol,  was  digested 
for  several  hours  in  a  reflux  apparatus  with  excess  of  methyl  iodide, 
and  the  methoxy  group  in  the  resulting  product  determined  by 
Zeisel's  method.  The  result  obtained  was  9-2  per  cent,  of  methoxyl , 
in  agreement  with  that  calculated  for  the  formula  C12H19O9OCH3. 

The  adsorption  of  basic  dyestuffs  by  cellulose  acetate. 

Certain  dyestuffs,  such  as  gentian  violet  are  adsorbed  in  con- 
siderable quantities  from  aqueous  solution  by  cellulose  acetate, 
the  solid  being  coloured  blue.  Cellulose,  it  is  true,  also  adsorbs 
the  dye,  but  to  a  much  smaller  extent,  and  the  solid  becomes 
violet.  This  property  may  be  utilized  to  identify  unaltered  cellulose 
in  commercial  preparations  of  cellulose  acetate.  Methyl  orange 
gave  negative  results,  but  methyl  red  was  adsorbed  in  considerable 
quantity,  the  solid  becoming  red.  Free  dimethylaminoazo benzene 
gave  negative  results,  but  the  hydrochloride  of  this  base  was 
strongly  adsorbed,  the  solid  cellulose  acetate  assuming  a  pinkish 
yellow  colour  and  the  colour  of  the  aqueous  solution  being  almost 
completely  discharged. 

The  authors  desire  to  express  their  grateful  thanks  to  Mr  J.  W. 
H.  Oldham,  M.A.,  of  Trinity  College,  for  much  valuable  assistance 
in  connexion  with  this  investigation.  Mr  Oldham  has  also  carried 
out  a  large  number  of  experiments  on  the  influence  of  the  mode  of 
preparation  upon  the  resulting  properties  of  cellulose  acetate,  and 
it  is  hoped  that  his  results  when  completed  may  form  the  subject 
of  a  future  communication. 


Mr  Molin,  An  examination  of  SearWs  method,  etc.         23 


An  examination  of  Searle's  method  for  determining  the  viscosity 
of  very  viscous  liquids.  By  Kurt  Molin,  Filosofie  Licentiat, 
Physical  Institute,  Technical  College,  Trondhjem.  (Communicated 
by  Dr  G.  F.  C.  Searle.) 

{Read  9  February  1920.] 

§  1.  The  determination  of  the  coefficient  of  internal  friction  in 
very  viscous  liquids  has  been  the  object  of  measurements  by  many 
different  methods.  A  review  of  these  will  be  found  in  Reiger*. 
A  number  of  more  recent  methods  are  given  by  Kohlrauschf,  and 
among  them  is  a  method  of  Searle'sJ.  An  examination  of  this 
method  is  the  object  of  the  present  paper. 

In  his  paper,  "A  simple  viscometer  for  very  viscous  liquids," 
Dr  SearleJ  gives  an  account  of  a  viscometer  he  has  constructed. 
The  method  consists  in  causing  a  vertical  cylinder  to  rotate  within 
a  coaxal  cylinder  containing  liquid,  and  in  determining  the  angular 
velocity  of  the  inner  cylinder  for  a  known  value  of  the  driving 
couple.  The  couple  is  produced  by  the  weights  of  two  loads  acting 
on  a  drum  by  two  threads.  The  time,  T  seconds,  of  one  revolution 
of  the  cylinder  is  found,  and  the  length,  I  cm.,  of  the  inner  cylinder 
immersed  in  the  liquid  is  observed. 

Newton's  statement  is  that 

f--^Tn'  <1) 

where/  is  the  force  per  unit  area  which  acts  against  the  direction 
of  motion  and  at  right  angles  to  the  normal,  n,  to  the  surface, 
dV/dn  is  the  velocity  gradient,  and  rj  is  the  coefficient  of  viscosity. 
In  this  statement  the  motion  of  the  liquid  is  supposed  to  take  place 
parallel  to  a  fixed  plane.  Treating  the  liquid  as  incompressible, 
and  modifying  (1),  by  substituting  the  rate  of  shearing  for  dV/dn, 
so  as  to  suit  the  case  of  rotation,  we  obtain  the  following  formula: 

gD  (a2  _  62)  fMT\      ^  (MT\ 

Here  D  is  the  effective  diameter  of  the  drum,  a  and  h  are  the  radii 
of  the  cylinders,  and  M  is  the  mass  of  each  of  the  two  loads,  which 
are  required  to  move  the  inner  cylinder  with  the  constant  angular 
velocity  Q,  such  that  2ttJQ.  =  T. 

*  R.  Reiger,  Ann.  d.  Phys.,  19,  p.  985,  1906. 

t  F.  Kohlrausch,  Lehrbuch  d.  praktischen  Physik,  xii.  Aufl.,  p.  268. 

%  G.  F.  C.  Searle,  Proc.  Cambridge  Phil.  Soc,  16,  p.  600,  1912. 


24  Mr  Molin,  An  examination  of  Searle's  method 

The  angular  velocity  of  the  liquid  about  the  axis  of  the  cylinders, 
at  a  distance  r  from  the  axis,  is  given  by 

_  277  62         /^  ^ 

^       T  '  a^-bAr^~ 

When  r  =  b,  the  radius  of  the  inner  rotating  cylinder, 

oj  =  1^  =  27r/r, 

and  when  r  =  a,  the  internal  radius  of  the  outer  fixed  cylinder, 
CO  =0.  This  problem  was  first  treated,  not  quite  accurately,  by 
Newton.  The  above  results  were  given  substantially  by  Stokes  *, 
and  are  also  given  by  Lambf  and  by  SearleJ. 

The  rate  of  shearing,  rdco/dr,  varies  somewhat  as  r  increases 
from  b  to  a,  as  is  shown  by  the  formula 

doi  27r         2a%^ 

r 


dr  T  '  (a2  _  §2)  ^2  • 

We  have  only  taken  into  account  the  friction  between  the 
coaxal  cylindrical  layers  of  the  liquid  and  not  the  friction  between 
the  horizontal  layers  in  proximity  to  the  bottom  surface  of  the 
movable  cylinder,  and  have  not  considered  the  conditions  that 
arise  near  that  surface.  In  practice,  only  the  lower  end  of  the 
rotating  cylinder  is  exposed  to  viscous  action ;  Dr  Searle  makes  an 
allowance  for  this  end  by  writing 

^^^•r+i'  (2) 

where  I  is  the  length  by  which  the  height,  /,  of  the  liquid,  in  the 
simple  theory,  must  be  increased,  in  order  that  the  increase  of 
couple  shall  correspond  to  the  viscous  action  in  proximity  to  the 
end  surface  and  the  edge  of  the  rotating  cylinder. 

Dr  Searle  gives  a  graphical  method  of  determining  k.  The 
values  of  MT  are  plotted  against  I,  and  he  says,  "It  will  be  found 
that  the  points  lie  on  a  straight  line,  which  cuts  the  axis  of  I  at 
a  distance  k  from  the  origin."  Dr  Searle  adds  "If  the  corresponding 
total  load  hung  from  each  thread  be  M  grammes,  it  will  be  found, 
on  repeating  the  observation  with  various  loads,  that  MT  is 
constant  for  a  given  level  of  liquid.  This  result  confirms  the 
fundamental  assumption  that  the  viscous  stress  at  each  point  is 
proportional  to  the  rate  of  shearing  of  the  liquid." 

*  G.  G.  Stokes,  Brit.  Ass.  Report,  p.  539,  1898. 
t  H.  Lamb,  Hydrodynamics,  Third  Ed.,  p.  546,  1906. 
loal  ^'  ^'  ^'  ®^*^^®'  ^°^-  ^^f-'  P-  602.  Compare  C.  Brodman,  Wied.  Ann.,  45,  p.  163, 


for  determining  the  viscosity  of  very  viscous  liquids         25 

§  2.  In  my  experiments  I  used  Dr  Searle's  viscometer,  as 
supplied  by  Messrs  W.  G.  Pye  and  Co.,  Cambridge*.  I  determined 
the  viscosity  of  treacle,  as  Dr  Searle  refers  to  a  determination  of  77 
for  that  liquid.  I  found  26  =  3-74  cm.,  2a  =  5-01  cm.,  and 
D  =  1-95  cm.  Since  g  =  982  cm.  sec. -2  at  Trondhjem,  the  con- 
stant C  has  the  value 

C  =  3-070  ±  0-035. 

From  the  data  given  by  Dr  Searle,  I  find  for  the  constant  of  the 
instrument  used  by  him,  Cg  =  3-153. 

In  my  instrument  the  rate  of  shearing  for  radius  r  is  given  by 

^_  _27r    15-80 
'^  dr~       T  '     r^    ' 

§  3.  To  examine  how  MT  depends  upon  M,  when  I  is  kept 
constant,  six  series  of  observations  were  taken  with  six  values  of  I 
varying  from  10-0  to  2-15  cm.,  and  in  each  series  M  was  made  to 
vary  from  5  to  205  grammes. 

Since  the  viscosity  of  highly  viscous  substances  diminishes  very 
rapidly  as  the  temperature  increases,  as  was  shown  by  Reigerf  and 
by  Glaser J  for  values  of  rj  of  the  magnitudes  4-8  x  10^  to  67-2  x  10®, 
and  by  Ladenburg§  for  '>7  =  1-3  x  10^,  great  care  must  be  taken 
to  keep  the  temperature  constant.  The  apparatus  was,  therefore, 
placed  in  a  thermostat  with  electric  temperature  regulation,  and 
a  very  constant  temperature  of  19-8°  C.  was  /naintained.  The 
apparatus  was  left  in  the  thermostat  for  24  hours  before  the 
measurements  were  begun,  and,  during  the  short  time  a  rotation 
trial  was  in  progress,  only  the  outer  wooden  door  of  the  thermostat 
was  opened,  since  one  could  see  into  the  thermostat  through  the 
inner  glass  door.  The  final  measurements  were  all  carried  out  in 
the  course  of  a  day;  the  observations  were  made  at  intervals  of 
about  10  minutes,  so  that  the  unavoidable  disturbances  of  tempera- 
ture, due  to  the  manipulations,  might  have  time  to  disappear. 

In  other  respects  the  measurements  were  carried  out  in  ac- 
cordance with  Dr  Searle's  II  instructions.  The  revolutions  were 
timed  by  aid  of  a  stop-watch  and  the  times  were  taken  for  different 
numbers  of  revolutions  with  odd  numbers  up  to  9,  as  well  as  the 
average  time  for  one  revolution.  As  no  decrease  in  the  time  of  a 
single  revolution  could  be  noticed  as  the  rotation  continued,  the 
divergences  from  the  mean  lying  within  the  limits  of  the  errors 

*  Catalogue  of  Scientific  Apparatus  manufactured  by  W.  G.  Pye  and  Co., 
List  No.  120,  p.  39,  1914. 

t  R.  Reiger,  loc.  cit.,  p.  998. 

X  H.  Glaser,  Ann.  d.  Phys.,  22,  p.  719,  1907. 

§  R.  Ladenburg,  Ann.  d.  Phys.,  22,  p.  309,  1907. 

II  G.  F.  C.  Searle,  loc  cit.,  p.  603. 


26  Mr  Molin,  An  examination  of  Searle's  method 


of  observation,  there  was  no  observable  acceleration.  AVe  may 
conclude  that,  even  for  the  greatest  values  of  M,  the  viscosity  of 
the  liquid  remained  sensibly  constant,  in  spite  of  the  fact  that 
some  potential  energy  was  converted  into  heat. 

The  values  of  T  found  in  these  experiments  are  given  in 
Table  1. 

Table  1. 

Time,  in  seconds,  of  one  revolution  of  cylinder. 


M 

Z  =  10-0 

Z  =  8-45 

Z=:7-65 

Z  =  5-50 

Z  =  3-30 

l  =  2-\5 

grm. 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

5 

129-0 

7 

114-6 

50-7 

10 

120-4 

108-7 

100-2 

71-3 

44-7 

12 

93-3 

85-0 

77-0 

54-3 

15 

71-5 

65-3 

59-0 

42-3 

26-6 

19-5 

20 

52-8 

47-0 

41-7 

29-7 

25 

4M 

35-9 

32-2 

23-6 

14-5 

11-1 

30 

33-3 

29-6 

26-4 

19-0 

35 

28-8 

24-8 

22-2 

16-3 

10-0 

40 

251 

21-7 

19-2 

14-0 

45 

22-0 

19-0 

171 

12-3 

55 

17-9 

15-2 

13-6 

101 

6-3 

4-6 

65 

14-9 

12-6 

11-4 

8-3 

75 

12-8 

10-9 

9-8 

7-2 

4-5 

3-4 

105 

9-1 

7-7 

7-0 

5-1 

155 

61 

5-2 

4-S 

3-5 

205 

4-6 

3-9 

3-6 

The  results  have  been  plotted  in  the  form  of  six  curves  each 
for  one  value  of  I,  as  in  Diagram  1.  The  curves  are  represented  in 
the  form  T  {MT,  M )^.eonst.  =  0. 

From  the  diagram  it  is  clear  that  the  function  T  {MT,  T)i  =  0 
does  not  represent  a  family  of  straight  lines  parallel  to  the  ilf-axis, 
and  that  each  of  the  six  curves  has  a  hyperbolic  appearance.  When 
M  approaches  a  certain  lower  limit  Mq,  MT  tends  to  infinity. 
The  area  covered  by  the  group  of  curves  can  be  divided  by  a 
parabolic  boundary  curve  into  two  departments,  in  one  of  which 
MT  is  sensibly  constant  for  a  given  value  of  I. 


§  4.    I  have,  further,  examined  how  MT  depends  upon  I,  when 
M  is  kept  constant,  and  have  found  that  the  function 

F  {MT,  Z)^,=eonst.  =  0 


for  determining  the  viscosity  of  very  viscous  liquids         27 


represents,  not  a  single  straight  line*,  but  a  family  of  approxi- 
mately straight  lines.  Each  line  can  be  represented  by  the  equation 
MT  =al  +  p.  For  this  group  of  curves  d  {MT)ldl  tends  to  a  definite 
value  as  M  increases,  i.e.  the  curves  approach  a  certain  border  line 


MT 

IhagTa; 

Til- 

I 

Wn;i 

i)-o. 

* 

t 

^ 

l^ 

'~~~~ 

\\M 

l=1QO 

1 

\ 

\ 

-^ 

/  / 

(1-5,3) 

T=a^5 

\ 

--- 

/   / 

1=765 

/     * 

V 

J 

(1-H.3) 

■^ 

r' 

il-3.'^] 

1=5,50 

- 

«K>0 

V 

il 

[' 

(i-:i,35) 

\ 

\ 

•v.,^^^/' 

]=S.SO 

\ 

/ 

\ 

■"--^ 

/ 

fl-hl 

1=Z.15 



/ 

/ 

/ 

N 

y 

Calculat 

e^forl- 

3. 

^ 

o      lO     10    30    it-o    so    €0    70    60    90    100  110   1Z0   r3o  i4o    ;5o  160  170  180  190  200  aw 

which  is  comparable  with  Searle's  straight  line.  The  coefficients 
a  and  ^  have  been  calculated  for  each  line  by  the  method  of  least 
squaresf ,  using  the  formulae 

1.1 .  HMT  -  6111 .  IslMT      ^      1.1 .  i:iMT  -  IIMT  .  ZV- 


(SZf  -  6S/2 


(S^)2  -  6SZ2 


*  G.  F.  C.  Searle,  loc.  cit.,  p.  604. 

t  F.  Kohlrausch,  Lehrbuch  d.  jn-aktischen  Physik,  p.  13,  1914. 


28 


Mr  Molin,  An  examination  of  SearWs  method 


the  various  observations  being  regarded  as  having  equal  weights. 
The  values  of  a  and  ^  have  been  thus  calculated  for  seven  different 
lines,  and  the  results  are  given  in  Table  2. 

Table  2. 
Values  of  a,  §  and  k. 


M  grm.     , 

a 

^ 

k  cm. 

12 

105-86 

78-40 

0-740 

15 

102-64 

65-61 

0-639 

20 

99-29 

53-54 

0-539 

35 

108-80 

47-79 

0-439 

65 

90-78 

43-83 

0-482 

75 

90-27 

43-63 

0-483 

105 

89-79 

45-47 

0-506 

1 


When  I  =  0,  then  MT  =  ^,  and  Table  2  shows  how  ^  varies  with  M. 
The  curve  thus  extrapolated  for  ^  =  0  is  marked  "Calculated  for 
Z  =  0"  in  Diagram  1. 

When  MT  =  0,  we  have  ^  =  |  ^  |  =  |  /3/a  |  ,  where  k  is  the 
correction  for  the  lower  end  of  the  rotating  cylinder. 


0,7 
0,6 

qs 

Q3 


\ 

Diagram  % 

^ 

\ 

2(k, 

n]=q 

'x 

o    • 

— o— 

— «— 

10     ZO      30     ^O      50      60     70      80     90     100    110     ISO     ISO    1^^o  I^  gr. 


Diagram  2  shows  how  k  depends  upon  M. 
The  facts  here  recorded  show  that  equation  (2)  should  be 
replaced  by 

M,T 


-n^c 


i  +  k,' 

where  k^  is  the  value  of  k  corresponding  to  the  load  M^. 


.(3) 


for  determining  the  viscosity  of  very  viscous  liquids         29 

If  the  value  of  h^  corresponding  to  M^  is  read  off  from  the 
curve  of  Diagram  2,  the  viscosity  17  can  be  calculated  by  equa- 
tion (3).  The  values  of  k  found  from  Diagram  2  have  been  used 
in  forming  Table  3. 

Table  3. 

Values  of  M^T/il  +  k^). 


M 

Z  =  lO-0 

i  =  8-45 

1  =  1 -m 

?  =  5-50 

Z  =  3-30 

grm. 

cm. 

cm. 

cm. 

cm. 

cm. 

10 

111-7 

118-0 

118-5 

]  13-4 

109-5 

12 

104-2 

111-0 

110-0 

104-5 

15 

101-2 

108-0 

106-8 

103-2 

1010 

20 

99-3 

104-5 

102-0 

98-6 

25 

97-9 

100-1 

99-1 

98-5 

95-1 

30 

96-5 

99-0 

97-2 

95-4 

35 

95-6 

97-0 

95-6 

94-7 

92-5 

40 

950 

970 

94-5 

93-7 

45 

94-5 

96-0 

94-3 

92-5 

55 

93-3 

94-0 

91-5 

92-4 

91-6 

65 

92-4 

92-0 

90-7 

89-7 

75 

92-0 

91-7 

90-4 

89-7 

105 

91-1 

90-2 

90-4 

901 

155 

90-3 

89-2 

90-6 

89-6 

205 

90-0 

89-3 

89-3 

From  Table  3  it  appears  that  the  area  in  Diagram  1  in  which 
equation  (3)  holds  good  is  restricted  to  that  part  of  the  diagram 
to  which  the  parabolic  boundary  curve  is  convex.  From  the  values 
of  MT  derived  from  Table  1  and  plotted  in  Diagram  1,  the  equation 
of  the  parabola  is  found  to  be  M'^  =  11-26  {MT).  I  have  not  been 
able  to  give  the  parabola  any  definite  physical  interpretation,  and 
it  ought  to  be  regarded  as  representing  a  diffuse  limit  region.  But 
it  is  only  when  we  pay  regard  to  this,  that  we  obtain  values  of  7) 
differing  from  each  other  by  amounts  lying  within  the  limits  of 
experimental  error*.  To  make  a  comparison  with  the  values  of 
M  and  I  which  Dr  Searle  has  used,  I  have,  in  Diagram  1,  plotted 
(the  broken  hne)  his  values  of  i/Tf  (strictly  speaking,  MTjC, 
which  are  comparable  in  magnitude  with  my  values  of  MT) 
against  M. 

Dr  Searle  has  pointed  out  to  me  that  the  effect  shown  in 
Diagram  1  might  conceivably  be  due  to  pivot  friction.  I  have 
carefully  considered  this  possibility.  Before  the  liquid  was  put 
into  the  apparatus,  I  adjusted  the  pivots  so  that  the  rotation  due 

*  Compare  G.  F.  C.  Searle,  loc.  cit.,  Table  II,  p.  606. 

I  Calculated  from  Table  1,  G.  F.  C.  Searle,  loc.  cit.,  p.  605. 


30 


Mr  Molin,  An  examination  of  Searle's  method 


to  the  weights  of  the  two  empty  pans  (5  grm.  each)  was  so  rapid 
that  I  was  hardly  able  to  measure,  for  instance,  3T  by  using  a 
stop  watch.  I  have,  therefore,  not  been  able  to  take  account  of 
any  pivot  friction.  This  cause  of  error  would,  at  any  rate,  produce 
effects  much  smaller  than  those  actually  found. 


§  5.    From  the  results  for  M  =  205  grm.  given  in  Table  1  we 
find  the  mean  value 

-q  =  TIA:-1  dyne  sec.  cm.-^, 

for  the  temperature  of  +  19-8°  C.    To  show  how  t]  depends  upon 


3^0 

(   3« 

^j 

1 

Diagrams.                        ja 

3« 

r 

4 

— 1— 

1 

- 

1 
1 

1 

0  S>«nt  1. 

ViS 

i 

* 

Vtt 

\ 

320 

— r^ 

\ 

3. 

\ 

■Q 

315 
310 
305 

io 

• 

—  i: nngular  veiociL^ oi  the                     "^-q 
Rotating  C^inder. 

qi     GX 

\ 

'\ 

« 

L     Tf(i3- 

90'r 

° 

\  J 

290 
285 
280 

•o\ 

A 

• 

\ 

« 

\ 

^j 

'        * 

.> 

^^ 

^ 

e 

e 

270 

■ 

, 

A 

0       0,1      Qi     03     0^      q5     Q«     Cr7      <;»     <^9       ^O      -p       V^      t^      y^      \5      1/5       ^7       1^ 

the  angular  velocity  O  =  27t/T,  the  values  of  17  and  Q,  obtained 
from  the  first  three  series,  have  been  plotted  in  Diagram  3.  The 
curve  drawn  among  the  plotted  points  suggests  that  the  relation 
between  -q  and  Q.  can  be  expressed  in  the  form 

7]  =  274-7  +  <f)  exp  (-  AO^). 

To  find  the  constants  cf),  A  and  x,  I  considered  the  equation 

log,  [rj  -  274-7)  -  log,^  -  A^^' (4) 


/or  determining  the  viscosity  of  very  viscous  liquids         31 

When  the  values  of  loge  (-7  —  274-7)  were  plotted  against  O,  the 
curve  was  roughly  a  straight  line.  Hence  x  may  be  taken  as  unity, 
and  thus  the  number  of  constants  to  be  found  is  reduced  to  two. 
By  the  method  of  least  squares,  I  obtained  logf<^  =  4-375  and 
A  =  5-694,  and  thus 


r^ig.g  =  274-7  +  79-44  e-^-''^'*". 


.(5) 


Equation  (5)  expresses  the  results  of  the  observations  when 
Q.  exceeds  0-1,  but  not  for  smaller  values  of  Q. 

§  6.  Experiments  carried  out  at  different  temperatures  showed 
that  the  curves  representing  the  function 


T  [MT,  M\ 


0 


are  of  the  same  character  as  those  given  in  Diagram  1.  Table  4 
gives  the  values  of  7]  found  for  various  temperatures.  In  these 
experiments  I  was  10-0  cm. ;  and,  at  each  temperature,  six  different 
loads  were  used,  in  order  that  I  might  be  able  to  decide  with 
certainty  that  the  values  of  M,  used  in  calculating  the  value  of  17 
for  each  temperature,  lay  in  the  area  to  the  right  of  the  parabolic 
boundary  line  of  Diagram  1.  The  same  value  of  k,  viz.  the  limiting 
value  0-48  cm.  shown  in  Diagram  2,  was  used  in  calculating  the 

Table  4. 
Values  of  7]  at  various  temperatures. 


Temp. 

V 

Temp. 

■n 

t°C. 

Dyne  sec.  cm."^ 

t°C. 

Dyne  sec,  cm.  ^ 

19-8 

274-7 

8-75 

1950 

18-0 

415 

6-2 

2700 

13-0 

860 

60 

2750 

11-8 

1140 

2-8 

4970 

11-6 

1200 

various  values  of  rj.  These  values  are  not  claimed  to  be  exact. 
In  these  experiments  it  was  very  difficult  to  keep  the  temperature 
constant  during  each  series  of  observations,  and  thus  a  deter- 
mination of  k  for  each  temperature  was  out  of  the  question.  From 
the  curve  of  the  function  -q  =f{t),  shown  in  Diagram  4,  it  follows 
that  I  drj/dt  I  rises  rapidly  as  -q  increases;  this  tallies  with  what 
was  said  above. 


"32  Mr  Molin,  An  examination  of  Searle's  metJiod 


4000 


9        10       Tl        1Z        13       IV       IS       IS       17       18       19       20 


jTemptX- 


§  7.  I  thought  it  would  be  interesting  to  compare  the  results 
given  by  Searle's  method  with  those  obtained  by  Poiseuille's 
method.  The  utility  of  the  latter  method  for  very  viscous  liquids* 
is  proved  by  the  investigations  of  Kahlbaum  and  Eaberf  for 
values  of  rj  in  the  neighbourhood  of  40,  and  by  LadenburgJ  for 
r]  =  1-3  X  10^.  Fausten§  has  found  that  the  length  of  the  dis- 
charge tube  must  exceed  45  cm.,  if  the  simple  Poiseuille  formula 

is  to  represent  actual  facts.  In  the  formula 

h  =  Height  of  liquid  corresponding  to  difierence  of  pressure 
between  ends  of  tube. 

R  =  Internal  radius  of  tube.  L  =  Length  of  tube. 

p  =  Density  of  liquid  (=  1-4103  ±  0-0003  grm.  cm.-^at  19-8°  C). 

m  =  Mass  of  liquid  discharged.  t  =  time  of  discharge. 

For  shorter  tubes,  Hagenbach's*  correction  must  be  employed; 
otherwise  the  value  obtained  for  77  will  be  too  high.  As  the  liquid 
flows  out  into  the  air  in  an  even  jet,  it  carries  kinetic  energy  with 
it;  in  order  to  allow  for  this,  the  value  of  t]  given  by  Poiseuille's. 

*  H.  Glaser,  Eriangen  Diss.,  1906. 

t  G.  W.  A.  Kahlbaum  and  S.  Raber,  Acta  Ac.  Leap.,  84,  p.  204,  1905. 

X  R.  Ladenburg,  Ann.  d.  Phys.,  22,  p.  298,  1907. 

§  A.  Fausten,  Bonn.  Diss.,  1906. 


for  determining  the  viscosity  of  very  viscous  liquids         33 

formula  must  be  multiplied,  according  to  Hagenbach*,  by  a  cor- 
recting factor  slightly  less  than  unity.  As  the  thermostat  could 
only  accommodate  tubes  shorter  than  45  cm.,  Hagenbach's  correc- 
tion was  calculated,  but  was  found  to  be  negligible.  Ladenburgf 
points  out  that  both  Hagenbach's  and  Couette's  corrections  to 
Poiseuille's  formula  can  be  entirely  ignored  for  liquids  such  that  t] 
is  of  the  magnitude  1-3  x  10^. 

The  discharge  vessel  consisted  of  a  wide  glass  cylinder;  through 
the  bottom  of  this  was  bored  a  hole  through  which  the  discharge 
tube  was  connected  with  the  interior  of  the  cylinder.  The  whole 
apparatus  was  placed  in  the  thermostat  and  the  same  temperature, 
19-8°  C,  was" maintained  as  was  used  in  the  earlier  experiments, 
AVhen  a  tube  whose  internal  radius  was  about  0-26  cm.  was  used, 
the  liquid  did  not  issue  in  a  continuous  jet  but  in  drops.  The 
values  obtained  for  -q  are  given  in  Table  5.  The  mean  value  is 
r]  =  271-1.  The  value  obtained  by  Searle's  method,  viz.  274-7, 
differs  from  that  obtained  by  Poiseuille's  method  by  1-3  per  cent.; 
the  agreement  may  be  regarded  as  good. 


Table  5. 
Values  of  7]  by  Poiseuille^s  method. 


Rem.. 

Zcm. 

h  cm. 

m  grm. 

t  sec. 

■n 

0-3168 

46-48 

49-36 
49-93 

54-421 
53-568 

1790 
1757 

269-9 
272-3 

§  8.    The  influence  of  the  base  of  the  rotating  cylinder  can  be 
eliminated,  without  determining  k,  by  using  the  relation  J 

77  =  C  . ^ -, =  Cy, 

n  ~  h 

provided  that  the  points  corresponding  to  M^T^  and  M^T^  He  to 
the  right  of  the  parabolic  boundary  line  in  Diagram  1.  If  we  put 
Zj  =  10-0  cm.,  we  obtain  the  results  given  in  Table  6. 


*  F.  Kohlrausch,  Lehrbuch  d.  praktischen  Physik,  pp.  264 — 269,  1914. 

t  R.  Ladenburg,  loc.  cit.,  p.  298. 

i  Compare  C.  Brodman,  loc.  cit.,  p.  163. 


VOL.   XX.   PART  L 


34         Mr  Molin,  An  examination  of  Searle's  method,  etc. 

Table  6. 
Values  of  y. 


MT 

944 

804 

734 

537 

342 

251 
215 

I 

10-0 

8-45 

7-65 

5-50 

3-30 

J 

90-2 

89-3 

90-5 

90-0 

88-4 

Wlien  the  various  values  are  given  the  same  weight,  the 
mean  value  of  y  is  89-7,  and  then  17  =  2754-. 

§  9.  Diagram  3  and  formula  (5)  show  that  77  cannot  be  re- 
garded as  independent  of  Q.  unless  Q.  exceed  a  certain  value,  in 
this  case  0-9.  Since  Q.  is  related  to  the  rate  of  shearing  rdw/dr, 
according  to  the  formula 


it  follows  that  7y  is  a  function  of  the  rate  of  shearing.  Hence,  the 
assumption  on  which  formula  (1)  is  based,  viz.  that  rj  is  independent 
of  the  rate  of  shearing,  seems  to  be  unjustifiable  for  small  values 
of  the  rate  of  shearing,  at  least  in  the  case  of  the  highly  viscous 
liquid  used  in  these  experiments. 


Miss  Haviland,  Note  on  Antennal  Variation  in  an  Aphis      35 


Preliminary  Note  on  Antennal  Variation  in  an  Aphis  (Myzus 
ribis,  Linn.).  By  Maud  D.  Haviland,  Fellow  of  Newnham 
College.    (Communicated  by  Mr  H.  H.  Brindley.) 

[Read  8  March  1920.J 

In  1918,  during  an  investigation  of  the  life-history  of  the  Red 
Currant  Aphis,  Myzus  ribis,  Linn.,  it  was  observed  that  consider- 
able variation  occurred  in  the  antennae  of  the  winged  partheno- 
genetic  females;  and  the  evidence  pointed  to  the  conclusion  that 
this  variation  was  induced  by  the  food^.  Antennal  variation  in 
certain  Aphididae  has  been  studied  by  Warren  ^  Kelly^  Ewing- 
and  Agar^.  Warren's  experiments  on  Hyalopterus  trirhodus 
showed  some  diminution  of  the  correlation  co-efficient  in  passing 
back  from  parent  to  grandparent.  Kelly,  for  Aphis  rumicis,  con- 
sidered that  somatic  variations  of  the  parents  were  not  inherited 
by  the  offspring.  Ewing,  who  bred  eighty-seven  generations  of 
Aphis  avenae,  concluded  that  the  variations  were  not  transmitted 
to  the  offspring. 

Agar  found  some  evidence  of  a  partial  inheritance  of  individual 
variations  in  Macrosiphum  antherini,  but  he  showed  that  this 
might  be  due  to  causes  other  than  true  inheritance. 

Myzus  ribis  is  a  common  pest  of  red  currant  bushes.  The 
sucking  of  the  aphides  upon  the  leaves  tends  to  cause  red  galls  or 
blisters,  within  which  the  plant  lice  continue  to  feed  and  reproduce. 
The  fifth  and  sixth  antennal  segments  of  the  winged  partheno- 
genetic  females  normally  bear  two  sense  organs  of  unknown 
function — one  on  the  distal  third  of  Seg.  v.,  the  other  on  the 
proximal  third  of  Seg.  vi.  It  was  observed  in  1918  that,  in  indivi- 
duals reared  on  red  blistered  leaves,  these  sensoria  were  placed 
comparatively  close  to  the  articulation  of  Segs.  v.  and  vi.  On  the 
other  hand,  if  the  aphides  were  fed  upon  green  unblistered  leaves, 
the  sensoria  were  placed  further  away  from  the  articulation. 

For  the  sake  of  brevity,  the  first  type  of  antenna  will  be  referred 
to  hereafter  as  the  Red  (or  R)  type,  and  the  second  as  the  Green 
(or  G)  type;  but  every  degree  of  transition  may  exist  between  the 
two  extreme  types. 

The  experiments  of  1918  were  incomplete,  and  were  conducted 
with  a  polyclonal  population.  They  were  repeated  in  1919  with  a 
monoclonal  population,  but  the  results  are  still  far  from  being- 
conclusive  owing  to  the  small  numbers  available  in  some  genera- 
tions. Only  the  winged  forms  show  the  required  character.  The 
production  of  these  forms  is  probably  governed  by  environmental 
factors  which  at  present  are  imperfectly  understood,  and,  for  some 

3—2 


36  Miss  Haviland,  Preliminary  Note  on  Antennal 

reason,  in  the  population  used  in  1919,  it  was  unusually  low.  It  is 
hoped  to  repeat  and  extend  the  range  of  the  experiments  in  1920. 
The  character  chosen  is  the  distance  between  the  sensoria  of 
antennal  segments  v.  and  vi.  and  the  articulation  of  these 
two  segments,  expressed  as  the  percentage  of  the  width  of  the 
head  between  the  eyes.  The  ratios  are  shown  separately  for 
each  segment,  with  a  dividing  line  to  represent  the  articulation. 

^,        ,„  ,       ,      ,,    ,  Seg.  VI.  =  19%  of  the  head- width 

Thus  J#  denotes  that  -^ ■^, — j— j — ^ — .  ,^,     . 

^  Seg.  V.  =  8%  of  the  head- width 

Each  generation  is  designated  by  combinations  of  two  letters: 
E,  (=  red  leaves)  and  G  (=  green  leaves)  and  numerals,  which 
express  its  complete  ancestry.  Thus  ^^^^  denotes  the  fourth 
generation  from  the  fundatrix  of  the  population,  and  the  F^. 
generation  after  transference  to  Green  leaves  after  two  consecutive 
generations  on  Red  blistered  leaves.  In  the  transferred  generations, 
the  aphides  were  removed  to  the  new  environment  when  less  than 
twelve  hours  old.  The  individuals  for  transference  were  selected 
wholly  at  haphazard.  Thus,  if  a  brood  mother  Eg  gave  birth  to 
four  young  in  the  day,  two  were  transferred  to  red  blistered  leaves, 
and  two  to  green  leaves,  and  so  on  in  equal  numbers  from  day  to  day. 

The  pure  Red  (RRR,  etc.)  lines,  and  pure  Green  (GGG)  lines 
were  used  as  controls.  The  latter  unfortunately  became  extinct  in 
the  third  (Gg)  generation.  Hence  for  later  generations  the  next 
longest  unbroken  line  on  green  leaves  (R2G0,  etc.)  had  perforce  to 
be  taken  as  the  control,  though  as  it  had  been  fed  for  the  first  two 
generations  upon  red  leaves,  it  cannot  be  regarded  as  wholly  * 
satisfactory.  In  Table  1,  the  curves  of  error  of  the  ratios  of  genera- 
tions R2,  R4  and  R2G4  are  shown.  Rg  is  the  common  ancestral 
generation.  The  mode  of  the  curve  of  R4  tends  to  shift  to  the  left, 
i.e.  the  ratios  of  the  antennal  segments  to  the  head-width  are 
smaller.  For  the  sake  of  clearness,  in  the  graph  only  the  curve  of 
R4  is  shown,  but  those  of  Rg,  R5  and  Rg,  though  with  a  smaller 
number  of  individuals,  are  almost  identical  with  it.  The  curves  of 
the  ratios  of  R2G1  and  R2G2  are  very  similar  to  their  red  controls. 
The  R2G3  generation  produced  very  few  winged  individuals,  but 
these  indicate  a  somewhat  greater  range  of  variation  in  Seg.  vi. 
The  curve  of  R2G4,  as  shown  in  the  graph,  has  a  marked  tendency 
to  shift  to  the  right,  indicating  that  the  ratio  of  the  antennal 
joints  to  head-width  has  increased,  and  this  tendency  is  maintained 
in  the  succeeding  generations,  R2G5  and  R2Gg.  The  position  in 
the  generation  series  does  not  account  for  the  change  in  the 
antennal  structure,  for  the  modes  for  the  six  Red  generations  are 
nearly  identical. 

So  far  we  have  considered  only  the  modes.  The  mean  ratios 
of  the  different  generations  are  dealt  with  in  the  succeeding  tables. 


I 


Variation  in  an  Aphis  (Myzus  ribis,  Linn.)  37 

Table  2  shows  the  mean  ratios  of  the  successive  generations  in 
four  lines  of  descent,  including  the  red  and  green  controls.  The 
extinction  of  the  green  control  line  was  unfortunate,  and  in  future 
experiments  it  will  be  very  desirable  to  obtain  a  pure  green  line. 
At  present  the  explanation  that  suggests  itself  of  the  variation  of 
the  RgGrg.  .  .  line  is  that  the  influence  of  red  feeding  persists  for  at 
least  two,  and  probably  three  generations  after  removal  to  different 
food,  and  this  is  somewhat  confirmed  by  the  R4G1.  .  .  etc.  line. 

Tables  3,  3a,  4,  4a  and  5,  5a,  give  the  effect  of  transference 
upon  the  mean  ratios  of  the  first,  second,  and  third  generations 
respectively,  and  below  each  is  an  analysis  of  the  ratio  of  each 
segment,  indicating  its  increase  or  decrease  over  previous  genera- 
tions and  the  controls. 

Examination  of  the  figures  seems  to  show  that  the  ratios  of  the 
first  generation  after  transference  vary  irrespectively  of  the 
parental  ratio. 

In  transference  to  Red,  the  ratio  of  Seg.  v.  increases  over  that 
of  the  parental  ratio,  but  in  Seg.  vi.  it  decreases  (Table  3).  In 
transference  to  Green,  the  results  for  both  segments  are  quite 
inconclusive  as  regards  the  parental  ratio  (Table  3a).  In  the 
second  generation  after  transference  to  Red,  the  results  are  like- 
wise inconclusive  for  both  segments  (Table  4).  After  transference 
to  Green,  the  ratio  of  Seg.  v.  shows  a  tendency  to  rise  above, 
and  Seg.  vi.  a  tendency  to  fall  below,  the  parental  and  grand- 
parental  ratios  (Table  4a). 

In  the  third  generation  after  transference  to  Red,  the  ratio  of 
Seg.  v.  rises  above  the  ancestral  ratios,  and  that  of  Seg.  vi.  falls 
(Table  5).  After  transference  to  Green,  the  ratio  of  Seg.  v.  rises 
above  those  of  the  ancestral  generations,  and  that  of  Seg.  vi.  rises 
in  one  case  and  falls  in  the  other  (Table  5a). 

These  results  are  inconclusive,  but  examination  of  the  control 
ratios  shows  that,  with  occasional  exceptions,  the  ratio  of  a  genera- 
tion with  a  mixed  ancestry  tends  to  rise  above  that  of  the  Red 
control,  but  remains  below  that  of  the  Green.  Many  more  experi- 
ments in  transference  are  required,  and  a  much  larger  number  of 
individuals  must  be  examined  before  any  conclusion  can  be 
reached;  but  at  present  the  evidence  suggests  that  the  antennae 
of  Myzus  ribis  are  modified  according  to  the  food  supplied,  and 
that  the  effect  induced  by  feeding  in  one  generation  is  discernible 
in  the  succeeding  three  or  four  generations.  It  is  difficult  otherwise 
to  explain  the  difference  between  the  ratios  of  Rg  and  R2G4,  and 
between  Rg  and  R4G3,  which,  translated  into  the  terms  of  human 
relationship,  would  be  third  cousins,  and  first  cousins  once  removed, 
respectively,  for  all  were  produced  by  parthenogenesis,  and, 
except  for  the  food,  reared  side  by  side  under  identical  environ- 
mental conditions. 


38 


Miss  Haviland,  Preliminary  Note  on  Antennal 


Table  1.  Curves  showing  the  ratio  of  the  distance  of  the  sensoria 
from  the  articulation  of  antennal  Segments  V  and  VI  to  the 
width  of  head.  The  lower  curves  refer  to  the  fifth,  and  the  upper 
to  the  sixth  segment. 


—  B2  generation 


=  E^Gi 


Table  2.    Mean  ratios  of  the  successive  generations  of  the  lines^ 


Cto 


i?3 ...,  JR/jG^ ...,  Hcfii  ...,  and  R2GQR1 


II 

III 

IV 

V 

VI 

VII 

VIII 

(^2¥ 

GsU 









R2  ^^0- 

R3¥- 

R4¥ 

R5  ¥- 

Re  -¥- 

— 

— 

)!         i) 

»         5> 

))    }■> 

R4G1  ""7" 

R4G2  V 

^iGsB 

— 

1)         H 

RA¥- 

E2G2 1§ 

R2G3  n 

R2G4  t"5 

R2G5  M 

RgGe  ft 

)>         )) 

>5                 ?) 

))     )) 

55            55 

R2G3R1  n 

R2G3R2  V 

— 

J 


Variation  in  an  Aphis  (Myzus  ribis,  Linn.) 


39 


Table  3.    Mean  ratios  of  the  first  generation  transferred  from  Green 
leaves  to  Red  blisters,  ivith  an  analysis  below. 
+   =  increase  over  ancestral  ratio 
-    =  decrease  from         „  ,, 

0   =  identical  with         ,,  ,, 


Generation 

Parental  Generation 

Red  Control 

Green  Control 

Ratio 

Ratio 

Ratio 

Ratio 

GiRi  -V- 

G  no  winged  forms 

R24? 

G2¥ 

G2R1  f* 

G,  ^^ 

R3¥ 

Gsfl 

R2GjRi  \f 

R2G1  ^^- 

R4¥ 

R2G2  i§ 

GlR2¥ 

GjRi  -gi- 

R3¥ 

G3ff 

R4GXR1  ¥ 

R4G1  ■\-- 

R6¥ 

R2G4  ^ 

R2G3R1  f§ 

R2G3  \i 

R6¥ 

R2G4  f  f 

R2G4R1  i\ 

^i^i  15 

Rc¥ 

R2G5  f  1 

Segment  V 


Generation 

Variation  from 

Parental 

Ratio 

Variation  from 

Red  Control 

Ratio 

Variation  from 

Green  Control 

Ratio 

GiRi 

no  winged  forms 

— 

0 

G2R1 

+ 

+ 

- 

R2GiRj 

+ 

+ 

- 

G1R2 

0 

0 

- 

R4GiRj 

+ 

0 

- 

R2G3R1 

0 

+ 

- 

R2G4Rj^ 

- 

+ 

— 

Segment  VI 

Generation 

Variation  from 

Parental 

Ratio 

Variation  from 

Red  Control 

Ratio 

Variation  froln 

Green  Control 

Ratio 

GiR, 

no  winged  forms 

+ 

- 

G2R1 

- 

+ 

- 

R2GxRi 

- 

0 

- 

G1R2 

- 

+ 

- 

R4GjRi 

- 

+ 

- 

R2G3R1 

0 

+ 

- 

R2G4R1 

— 

+ 

— 

40 


Miss  Haviland,  Preliminary  Note  on  Antennal 


Table  3a.    Mean  ratios  of  the  first  generation  transferred  from  Red 
blisters  to  Green  leaves,  with  analysis  as  in  Table  3. 


Generation 
Eatio 

Parental  Generation 
Eatio 

Green  Control 
Eatio 

Eed  Control 
Eatio 

ExGi  \«- 

R  no  winged  forms 

G2¥ 

R,^^ 

R2G1  ¥ 

R2  ¥ 

G-3  ft 

R3¥ 

GiRiGi  -V- 

GfiRi  "V" 

Gaff 

R3¥ 

RsGi^ 

R3  -¥ 

-       G3  f  f 

E4  ¥ 

GiRA¥ 

G1R2  ¥ 

R2G2 1§ 

E4¥ 

R4G1  ¥ 

R4  ¥ 

K2G3  15 

1 

Segment  V 


Generation 

Variation  from 

Parental 

Eatio 

Variation  from 

Green  Control 

Eatio 

Variation  from 

Eed  Control 

Eatio 

RiGi 

no  winged  forms 

0 

— 

R2G1 

- 

- 

- 

GiRiGi 

+ 

- 

+ 

R3G1 

0 

- 

0 

GjR2Gi 

+ 

- 

+ 

R4G1 

- 

- 

- 

Segment  VI 


Generation 

Variation  from 

Parental 

Eatio 

Variation  from 

Green  Control 

Eatio 

Variation  from 

Eed  Control 

Eatio 

RjGi 

no  winged  forms 

_ 

_ 

R2G1 

- 

- 

+ 

GjRiGx 

- 

- 

+ 

R3G1 

0 

- 

+ 

G^RgGj 

0 

+ 

+ 

R4(]ri 

+ 

+ 

+ 

Variation  in  an  Aphis  (Myzus  ribis,  Linn.) 


41 


Table  4.    Mean  ratios  of  the  second  generation  after  transference 
from  Green  leaves  to  Red  blisters,  with  analysis  as  in  Table  3. 


Generation 
Ratio 

Parental 

Generation 

Ratio 

Grand-parental 

Generation 

Ratio 

Red 

Control 

Ratio 

Green 

Control 

Ratio 

G,R2  V- 
G2R2  if 
R2G3R2  V- 

GiRi  -^ 
G2Ri  U 
R2G3R1  f  g 

Gj  no  winged  forois 

G2  ¥ 
R2G3  H 

R3¥ 

Rg¥ 

G3tt 
G3  If 
K2G5  f  § 

Segment  V 


Generation 

Variation  from 

Parental 

Ratio 

Variation  from 

Grand-parental 

Ratio 

Variation  from 

Red  Control 

Ratio 

Variation  from 

Green  Control 

Ratio 

G,R2 
G2R2 
R2G3R2 

0 

+ 

no  winged  forms 

+ 

0 

+ 

— 

Segment  VI 


Generation 

Variation  from 

Parental 

Ratio 

Variation  from 

Grand-parental 

Ratio 

Variation  from 

Red  Control 

Ratio 

Variation  from 

Green  Control 

Ratio 

(hR2 
R2G3R2 

— 

no  winged  forms 
0 

+ 
+ 

— 

42 


Miss  Haviland,  Preliminary  Note  on  Antennal 


Table  ia.    Mean  ratios  of  the  second  generation  after  transference 
from  Red  blisters  to  Green  leaves,  with  analysis  as  in  Table  3. 


Geueratiop 
Ratio 

Parental 

Generation 

Ratio 

Grand-parental 

Generation 

Ratio 

Green 

Control 

Ratio 

Red 

Control 

Ratio 

R2G2  i§ 
R4G2  \^ 

R,Gi  -V- 
R4G1  '^-f- 

R2  4P 
R4  V- 

G3  ti 
R2G4  ft 

Segment  V 

Generation 

Variation  from 

Parental 

Ratio 

Variation  from 

Grand-parental 

Ratio 

Variation  from 

Green  Control 

Ratio 

Variation  from 

Red  Control 

Ratio 

R2G2 
R4G2 

+ 
+ 

+ 
+ 

- 

+ 

+ 

Segment  VI 


Generation 

Variation  from 

Parental 

Ratio 

Variation  from 

Grand-parental 

Ratio 

Variation  from 

Green  Control 

Ratio 

Variation  from 

Red  Control 

Ratio 

R2G2 
R4G2 

- 

+ 

- 

+ 
+ 

Variation  in  an  Aphis  (Myziis  ribis,  Linn.) 


43 


Table  5.    Mean  ratios  of  the  third  generation  after  transference  from 
Green  leaves  to  Red  blisters,  with  analysis  as  in  Table  3. 


Generation 
Ratio 

Parental 

Generation 

Ratio 

Grand-parental 

Generation 

Ratio 

Great-grand - 

parental 

Generation 

Ratio 

Red 

Control 

Ratio 

Green 

Control 

Ratio 

G1R3  ¥ 

G1R2  ^' 

G,R,  i^i 

110  winged 
forms 

K4V- 

GsM 

Segment  V 


Generation 

Variation 

from 
Parental 

Ratio 

Variation 

from 

Grand-parental 

Ratio 

Variation  from 
Great-grand- 
parental 
Ratio 

Variation 

from  Red 

Control 

Ratio 

Variation 

from  Green 

Control 

Ratio 

G1R3 

+ 

+ 

no  winged 
forms 

+ 

- 

Segment  VI 


Generation 

Variation 
from 

Parental 
Ratio 

Variation 

from 

Grand-parental 

Ratio 

1 
Variation  from  !  Variation 
Great-grand-    '  from  Red 
parental            Control 
Ratio               Ratio 

Variation 

from  Green 

Control 

Ratio 

G1R3 

- 

- 

1 
no  winged             + 
forms 

1 

- 

44     Miss  Haviland,  Note  on  Antennal  Variation  in  an  Aphis 

Table  5a.    Mean  ratios  of  the  third  generation  after  transference 
from  Red  blisters  to  Green  leaves,  with  analysis  as  in  Table  3. 


Generation 
Ratio 

Parental 

Generation 

Ratio 

Grand- 

parental 

Generation 

Ratio 

Great-grand- 
parental 
Generation 
Ratio 

Green 
Control 
Ratio 

Red 

Control 

Ratio 

E2G3  u 
K4G3  !l 

R2G2  \% 
R4G2  -V- 

R2G1  -v- 

R4G1  5^ 

R4  V 

Gsff 
RsGsfl 

R5  ¥ 
Rg¥ 

Segment  V 


Generation 

Variation 

from 
Parental 

Ratio 

Variation 

from 

Grand-parental 

Ratio 

Variation  from 
Great-grand- 
parental 
Ratio 

Variation 

from  Green 

Control 

Ratio 

Variation 

from  Red 

Control 

Ratio 

R2G3 
R4G3 

0 

+ 

+ 
+ 

+ 
+ 

- 

+ 
+ 

Segment  VI 


Generation 

Variation 

from 
Parental 

Ratio 

Variation 

from 

Grand-parental 

Ratio 

Variation  from 
Great-grand- 
parental 
Ratio 

Variation 

from  Green 

Control 

Ratio 

Variation 

from  Red 

Control 

Ratio 

R2G3 

— 

— 

— 

— 

+ 

R4G3 

+ 

+ 

+ 

+ 

+ 

LITERATURE  REFERRED  TO  IN  THE  TEXT. 

(1)  Agar,  W.  E.  (1914).  "Experiments  on  Inheritance  in  Parthenogenesis," 
Phil.  Trans.  Roy.  Soc,  Series  B,  vol.  ccv,  pp.  421-487. 

(2)  EwiNG,  H.  E.  (1916).  "  Eighty-seven  generations  in  a  parthenogenetic 
pure  line  of  plant  lice,"  Biol.  Bull.,  vol.  xxxi,  No.  2,  pp.  53-112. 

(3)  Haviland,  Maud  D.  (1919).  "  On  the  Life  History  and  Bionomics  of 
Myzus  rihis,  Linn.,"  Proc.  Roy.  Soc.  Edinburgh,  vol.  xxxix,  pt.  1  (No.  8), 
pp.  78-112. 

(4)  Kelly,  J.  P.  (1913).  "Heredity  in  a  Parthenogenetic  Insect,"  Ainer. 
Nat.,  vol.  XLVii,  pp.  227-234. 

(5)  Warren,  E.  (1901).  "Variation  and  Inheritance  in  the  Parthenogenetic 
Generations  of  an  Aphis,  Hyaloptenis  trirhodus,  Walk.,"  Biometrika, 
vol.  I. 


Mr  Waran,  Ejfect  of  magnetic  field  on  Intensity  of  spectrum  lines  45 


The  effect  of  a  magnetic  field  on  the  Intensity  of  spectrum  lines. 
By  H.  P.  Waran,  M.A.,  Government  Scholar  of  the  University  of 
Madras.    (Communicated  by  Professor  Sir  Ernest  Rutherford.) 

[Read  8  March  1920.] 

[Plates  I  and  II.] 

Since  the  discovery  of  the  Zeeman  effect  the  main  attention 
has  been  directed  to  the  detailed  study  of  the  phenomenon  of  the 
small  change  of  wave  length  suffered  by  a  monochromatic  radiation 
in  a  magnetic  field.  The  question  whether  a  magnetic  field  affects 
the  spectrum  as  a  whole  has  not  received  much  attention. 

While  wiorking  on  the  Zeeman  effect  with  a  mercury  discharge 
tube  run  by  an  induction  coil  as  the  source,  a  small  portion  of  the 
capillary  tube  being  subjected  to  a  magnetic  field  of  about  5000 
c.G.s.  units  as  shown  in  Fig.  1,  the  light  was  observed  to  suffer  a 
change  in  intensity  and  also  in  colour  opposite  the  pole  pieces 
when  the  field  was  thrown  on.  A  spectroscopic  examination  revealed 
the  existence  of  some  selective  changes  in  the  spectrum  in  addition 
to  the  increased  brilliancy  of  the  general  spectrum.  It  was  also 
noticed  that  the  changes  taking  place  varied  considerably  with 
the  pressure,  at  a  low  pressure  the  tube  showing  little  change 
visually  but  greater  changes  in  the  general  spectrum.  Attention 
was  concentrated  on  the  latter. 

In  the  case  of  mercury  which  was  the  first  spectrum  investigated, 
the  tube,  containing  a  trace  of  residual  air  at  very  low  pressure, 
gave  the  principal  mercury  lines,  viz. : 

5790-66,  5769-6,  5460-7,  4916-0,  4358-34 
and  the  principal  hydrogen  lines 

6563,  4861-5  and  4340-7. 

On  applying  the  magnetic  field,  however,  marked  changes  were 
observed,  including  a  new  set  of  lines  at 

5426,  5679,  5872  and  5889, 
and  a  very  strong  red  line  at  6152,  brought  out  prominently  by 
the  field.  Mercury  lines  have  been  recorded  at  these  wave  lengths 
and  these  lines  brought  out  are  probably  due  to  mercury.  The 
behaviour  of  the  line  6152  was  very  remarkable.  It  was  invisible 
under  ordinary  conditions  but  showed  up  brilliantly  in  the  magnetic 
field,  the  effect  being  practically  instantaneous.  Exhausting  the 
tube  still  further  and  increasing  the  current  through  the  tube  to 
about  5  m.a.  Four  faint  lines  appeared  at  wave  lengths 
6234,  6152,  6123  and  6072, 


46 


Mr  Waran,  The  effect  of  a  magnetic  field 


and  corresponding  to  these  wave  lengths  mercury  lines  are  recorded 
by*  Stiles,  Eder,  Valenta,  Arons  and  Hermann.     But  Arons  and 


Fig.  1. 

Hermann  have  not  recorded  the  line  6152,  while  Stiles  records  it 
as  of  equal  intensity  with  the  line  6234.  Eder  and  Valenta  have 
not  observed  the  latter  lines  at  all,  but  record  the  line  6152  as 

*  Kayser,  Handbuch  der  Spectroscopie,  Band  v.  p.  538. 


on  the  Intensity  of  spectrum  lines  47 

one  of  veiy  great  intensity.  Examining  the  efEect  of  the  magnetic 
fields  on  these  four  lines,  it  is  very  interesting  to  note  that  the 
line  6152  alone  increases  about  five  times  in  brilliancy  while  the 
others  if  they  suffer  any  change  at  all,  decrease  in  intensity.  It  is 
also  interesting  to  note  that  this  line  6152  seems  to  be  the  same 
line  that  becomes  so  greatly  enhanced  when  the  tube  contains  a 
trace  of  helium  as  observed  by*  Collie.  It  seems  very  difl&cult  to 
excite  this  line  unless  at  least  a  trace  of  helium  is  present  in  the 
ajjparatus  and  at  this  stage  it  is  not  possible  to  suggest  any 
explanation  of  its  abnormal  behaviour. 

In  addition  to  these  very  prominent  changes  there  are  also 
many  minor  changes,  among  which  is  the  disappearance  of  a  faint 
trace  of  continuous  spectrum,  as  well  as  of  some  of  the  nebulous 
bands  and  lines,  the  remaining  lines  being  quite  sharp  on  a  dark 
background. 

The  abnormal  behaviour  of  the  mercury  spectrum  in  the  visible 
region  (the  ultra  violet  spectrum  has  not  yet  been  investigated) 
'suggested  the  study  of  other  spectra  and  the  spectrum  of  helium 
was  next  examined. 

The  discharge  tube  contained  hydrogen  and  a  slight  trace  of 
mercury  vapour  as  impurity  and  the  hydrogen  lines  and  the 
prominent  mercury  lines  were  also  visible.  The  effect  of  the 
magnetic  field  in  this  case  was  to  enhance  the  helium  lines  very 
considerably,  leaving  the  hydrogen  lines  practically  unaffected  or 
even  slightly  reduced  in  intensity.  In  this  spectrum  there  were 
also  a  few  faint  lines  not  yet  identified  definitely  which  remain 
quite  unaffected  by  the  magnetic  field.  In  the  further  study  of 
the  helium  spectrum,  the  gas  was  contained  in  a  separate  tube 
from  which  any  small  quantity  of  it  could  be  introduced  into  the 
discharge  tube.  At  a  pressure  of  1  mm.  of  mercury  the  addition 
of  a  small  trace  of  helium  produced  no  perceptible  effect  on  the 
spectrum  of  residual  air  which  showed  the  prominent  hydrogen 
lines  and  the  nitrogen  bands,  but  no  trace  of  any  of  the  helium 
lines.  But  on  switching  on  the  magnetic  field,  the  helium  lines 
flashed  out  prominently  and  disappeared  again  as  soon  as  the  field 
was  turned  off.  The  effect  is  shown  in  the  accompanying  photo- 
graphs (Plates  I  and  II).  In  a  plate  taken  with  a  greater  percentage 
of  helium  the  lines  are  visible  without  the  magnetic  field,  but  a 
great  enhancement  of  these  lines  with  the  field  is  evident,  and  a 
dense  new  fine  at  49334  a.u.  is  also  noticed  which  has  not  yet 
been  definitely  identified. 

The  spectrum  of  neon  was  also  studied,  and  in  a  tube  kindly 

lent  to  me  by  Dr  Aston,  there  was  a  trace  of  hydrogen  also  present, 

showing  the  three  principal  hydrogen  lines.    Here  also  the  effect 

of  the  field  was  to  enhance  very  considerably  the  neon  lines, 

*  Proc.  Roy.  Soc.  71,  25,  1902. 


48  Mr  Waran,  The  effect  of  a  magnetic  field 

leaving  the  hydrogen  lines  comparatively  unaffected,  so  that  by  a 
casual  examination  of  the  spectrum  the  hydrogen  and  the  neon 
lines  can  be  distinguished  from  one  another. 

The  oxygen  spectrum  is  rather  difficult  to  excite  when  mixed 
with  other  gases.  Yet  a  mixture  of  hydrogen,  oxygen  and  a  trace 
of  helium  was  tried  with  success  and  here  again  the  monatomic 
helium  lines  were  brouglt  out  by  the  magnetic  field,  leaving  the 
diatomic  oxygen  and  hydrogen  lines  comparatively  unaffected  as 
shown  in  the  photographs. 

From  these  experiments  the  natural  inference  follows  that  in 
a  mixture  of  the  monatomic  and  diatomic  gases,  the  monatomic 
gases  alone  seem  to  be  selectively  affected  in  a  peculiar  way 
resulting  in  their  spectrum  lines  alone  being  very  considerably 
enhanced  or  brought  out  prominently  even  when  not  visible  at 
all  previously.  By  this  method  minute  traces  of  the  monatomic 
gases  when  mixed  with  other  diatomic  gases  can  be  detected. 
On  this  view  we  might  also  explain  the  abnormal  mercury  line 
6152  and  others  as  due  to  the  radiation  from  the  monatomic 
atom  while  the  other  lines  may  be  classified  as  belonging  to  the 
molecule. 

Examining  the  spectrum  of  the  atmospheric  air  at  low  pressure 
in  this  way  the  effect  of  the  magnetic  field  is  to  bring  out  new  lines 
which  are  not  present  without  the  magnetic  field,  as  shown  in  the 
photographs.  As  far  as  their  wave  lengths  have  been  determined, 
though  one  or  two  of  them  fit  in  fairly  well  with  lines  catalogued 
as  belonging  to  oxygen  and  nitrogen,  yet  there  are  others  which 
are  difficult  to  identify  while  the  absence  of  other  stronger  lines  of 
oxygen  and  nitrogen  make  even  these  two  or  three  fits  inconclusive. 

Another  interesting  point  noted  in  these  experiments  is  the 
varying  degrees  of  enhancement  under  the  influence  of  the  field 
for  lines  belonging  to  the  same  element  helium.  Preston  has 
shown  that  the  Zeeman  effect  is  of  the  same  magnitude  for  lines 
belonging  to  the  same  series,  but  differs  in  different  series.  Simi- 
larly we  might  expect  the  degree  of  enhancement  of  the  lines  in 
the  magnetic  field  to  depend  on  the  series  to  which  the  line  belongs. 

The  exact  nature  of  this  phenomena  and  the  mechanism  of  the 
reaction  that  brings  about  these  novel  changes  in  the  general 
spectrum  is  not  yet  definitely  known  and  it  is  not  desirable  to 
attempt  an  explanation  until  the  study  of  the  spectrum  has  been 
extended  to  the  ultra  violet. 

The  current  in  the  tube  was  usually  about  3  m.a.  and  the  effect 
of  the  field  was  to  decrease  the  current  by  about  20  to  30  per  cent. 
The  changes  of  intensity  observed  cannot  be  attributed  to  this 
since  the  reduction  of  the  current  by  a  spark  gap  in  series  only 
brings  about  a  proportionate  decrease  in  brilliancy  of  the  general 
spectrum. 


Phil.  Soc.  Proc.  Vol.  xx.  Pt.  i 


(a)  Hydrogen  and 
Mercury 


(6)  Hydrogen  and 
Helium 


(c)  Hydrogen  and 
Helium,  low 
percentage 


(d)  Hydrogen  and 
Helium  in 
larger  per- 
centage 


(e)  Neon 


On 


Oft 


(/)  Air 


Ullltl    I    t   K    II 


On 
Off 


(gr)  Air 


On 


Off 


Fig.  2.  Photographs  showing  the  enhancing  effect  of  the  field.  The  small  lateral 
shift  is  due  to  the  camera  slider,  and  in  («)  the  mercury  line  6152  is  incUcated 
by  the  dot,  while  in  the  other  cases  the  lines  that  newly  turn  up  are  indicated 
by  the  arrows. 


I 


Phil.  Soc.  Proc.  Vol.  xx.  Pt.  i. 


Plate  II 


O      H, 


(1)  Air,  Oxygen,  Hydrogen 
and  trace  of  Helium 


(2)  Oxygen,  Hydrogen  and 
trace  of  Helium 


(3)  Excess  of  Helium  and 
trace  of  Air  and  Oxy- 
gen 


On 


OfE 


(4)  Oxygen,  Hydrogen  and 
Helium  and  trace  of 
Air 


On 


Off 


Fig.  3.     Photographs  showing  the  effect  in  mixtures  of  gases  studied. 


on  the  Intensity  of  spectrum  lines  49 

It  may  be  of  interest  to  note  that  in  solar  spectroscopy  the 
spectrum  of  the  sunspots  is  found  to  difTer  in  many  respects  from 
that  of  the  photosphere,  considerable  numbers  of  enhanced  lines 
occurring  in  the  sunspot  spectrum.  The  existence  of  a  powerful 
magnetic  field  in  sunspots  has  been  demonstrated  by  the  Zeeman 
effect  and  possibly  the  differences  in  the  spectrum  of  the  sunspot 
and  the  photosphere  may  be  attributed  to  this  new  effect  of  the 
magnetic  field  on  the  spectrum. 

The  further  study  of  this  effect  and  the  examination  of  other 
spectra  are  in  progress. 

Cavendish  Laboratory, 
Cambridge. 


VOL.   XX.   PART  I. 


50  Mr  Brindley,  Further  Notes  on  the  Food  Plants 


Further  Notes  on  the  Food  Plants  of  the  Common  Earwig  (For- 
ficula  aiiricularia).   By  H.  H.  Brindley,  M.A.,  St  John's  College. 

[Read  8  March  1920.] 

In  a  paper  pubHshed  in  the  Proceedings  of  the  Cambridge  Philo- 
sophical Society,  xix,  Part  4,  July  1918,  p.  170,  I  recorded  certain 
observations  in  August  and  September,  1917,  on  the  food  plants  of 
the  Common  Earwig,  with  the  view  of  obtaining  more  exact  infor- 
mation than  was  then  available  as  to  the  damage  likely  to  be 
done  by  this  species  in  a  flower  or  kitchen  garden.  The  paper  also 
epitomised  recent  literature  on  the  subject,  a  consideration  of 
which  had  revealed  a  considerable  amount  of  diversity  and  want 
of  exact  information  as  to  the  favourite  food  plants  of  earmgs  in 
the  British  Isles.  The  observations  made  by  myself  were  on  earwigs 
kept  in  captivity  in  connection  with  a  statistical  enquiry  as  to  the 
variation  of  the  forcipes  which  is  still  in  progress.  The  observations 
in  1917  were  on  earwigs  from  St  Mary's,  Isles  of  Scilly,  and  those 
recorded  in  the  present  paper  were  made  in  the  second  half  of  the 
year  1918  on  a  collection  from  the  Bass  Rock,  which  swarms  with 
earwigs.  The  animals  were  all  adults  and  were  kept  in  large  glass 
dishes  bedded  with  sand  slightly  damped  occasionally.  Earwigs  re- 
main healthy  in  a  soaked  substratum  if  the  ventilation  is  good,  but 
in  captivity  in  a  warm  room  without  circulation  of  air  they  suffer 
heavy  mortality  from  fungoid  attack,  as  I  have  already  recorded 
{Proc.  Camb.  Phil.  Soc,  xvii,  Part  4,  Feb.  1914,  pp.  335-338).  The 
fungus  appears  to  be  usually  Entomofhthora  forflculae  (Picard, 
BuU.  Soc.  Etude  Vidg.  Zool.  Agric.  Bordeaux,  Jan.-Apr.,  1914, 
pp.  25,  37,  62).  The  importance  of  ventilation  and  of  normal  tem- 
perature is  well  illustrated  by  the  far  fewer  fungoid  attacks  and 
the  low  mortality  when  the  new  Insect  House  belonging  to  the 
Cambridge  Zoological  Laboratory  became  available  in  1919.  It  is 
at  present  too  early  to  say  how  far  an  improvement  is  obtainable 
in  the  survival  of  eggs  and  young  which  it  is  hoped  to  rear  in  the 
spring  in  normal  outside  temperatures  in  the  Insect  House.  Earwigs 
offer  a  great  contrast  to  cockroaches  as  regards  desire  for  water; 
the  latter  thrive  in  captivity  for  months  in  ^  warm  room  on  food 
which  is  entirely  dry,  while  earwigs  certainly  visit  water  to  drink, 
as  I  have  seen  in  both  the  captive  and  wild  conditions.  I  have 
previously  recorded  {Proc.  Zool.  Soc.  Lond.,  Nov.  1897,  p.  913) 
how  Stylofyga  orientalis  in  captivity  seems  to  pay  no  attention  to 
a  damp  sponge  when  that  is  the  only  source  of  moisture.  We  have 
however  to  bear  in  mind  that  the  Common  Cockroach  is  probably  an 


of  the  Common  Earwig  (Forficula  auricularia)  51 

immigrant  from  warmer  countries  of  the  East.  The  earwigs  under 
observation  during  the  past  three  summers  had  no  animal  food  save 
that  afforded  by  those  which  died.  In  order  to  obtain  information 
as  to  preference  for  one  kind  of  plant  above  another  they  were 
usually  given  three  different  species,  taken  haphazard,  at  a  time, 
for  a  period  of  two  or  more  days. 

In  the  following  summary  the  observations  of  1917  and  1918, 
with  a  few  made  in  1919,  are  combined.  The  dates  when  the  different 
foods  were  given  are  noted,  as  in  the  latter  part  of  September,  when 
the  animals  tend  to  become  lethargic,  and  in  the  succeeding  two 
months  the  desire  for  food  is  much  lessened,  even  in  the  artificial 
temperature  of  a  laboratory.  The  capital  letters  after  the  names  of 
the  plants  indicate  those  which  were  given  at  the  same  time,  and 
the  numbers  appended  indicate  the  preference  exhibited  by  the 
earwigs:  e.g.  in  food  group  M,  M^  was  attacked  more  than  M^,  M^ 
more  than  M^;  in  group  F,  F^  after  two  plants  indicates  that  they 
seemed  to  be  attacked  equally,  and  more  readily  than  F^:  while 
in  group  Q,  Q*'  indicates  that  the  plant  offered  was  not  attacked  at 
all.    Similarly  for  the  other  groups. 

24-26  Aug.  '17.    Alkanet,  Blue  {AncJmsa  sp.)  C-:  leaves  not  attacked;  petals 

gnawed  considerably. 
27-29  Aug.  '17.    Anemone,  White  Japanese  {Anemone  japonica)  D^:  leaves 

not  attacked ;  petals  eaten  moderately. 
1-23  Sept.  '17.   Apple  {Pyrus  Mains)  F^:  rather  unripe  fruit  with  skin  whole 

was  not  attacked;  but  when  cut  across  was  gnawed  moderately:  24-28 

Sept.  '18,  leaves  holed. 
24-28  Sept.  '18.   Ai-tichoke,  Jerusalem  {Helianthus  tuberosus)  W:  leaves  holed 

and  edges  gnawed  down  to  midrib;  tuber,  cross  sUce  attacked  vigorously 

and  its  buds  also  devoured. 
20  Sept.-5  Oct.,  3-17  Nov.  '18.  Asparagus  {Asparagus  officmalis)  OS  T":  leaves 

gnawed  a  little ;  fruit  not  attacked. 
26-31  Aug.  '18.    Aster,  Mauve  China  {Callistephics  chinensis)  K-:  leaves  not 

attacked ;  petals  and  flower  buds  much  eaten. 
6-11   Sept.   '18.    Aster,  Pink  China  {Callistephus  chinensis):   leaves  shghtly 

nibbled;  petals  much  eaten ; ^OM;ers  used  as  a  refuge. 
15-20  Sept.  '18.  Balm,  Pale  Mauve  {Melissa  officinalis)  J^ :  leaves  not  attacked : 

petals  of  buds  devoured. 
22-23  Aug.  '17.    Bean,  Dwarf  {Phaseolus  vulgaris)  B^:  leaves  nibbled  ver;^ 

shghtly. 
30-31   Aug.   '17.    Bean,  Scarlet  Runner  {Phaseolus  multiflorus)  E^:   leaves, 

floivers  and  pods  apparently  neglected:  16-18  Oct.  '18,  leaves  holed  a  good 

deal  and  edges  gnawed  down  to  veins. 
20-28  Oct.  '18.   Beard  Tongue,  Scarlet  {Pentstemon  sp.)  R":  leaves  and  flowers 

not  attacked. 
22-23  Aug.  '17.   Beet  {Beta  vulgaris)  B^:  leaves  much  attacked,  especially  the 

petioles,  which  were  opened  out  and  their  pith  devoured. 
20-24  Sept.  '18.    Bell  Flower,  White  {Campanula  sp.)  K^:  leaves  not  touched; 

petals  completely  devoured. 
31  Aug.-6  Sept.  '18.  Bindweed,  Common  {Convolvulus  sp.):  leaves  much  holed. 
11-13  Sept.  '18.   Blackberry  {Rubus  fruticosus):  vipe  fruit  well  gnawed. 

4—2 


52  Mr  Brindley,  Further  Notes  on  the  Food  Plants 

30-31  Aug.  '17.  Cabbage,  Garden  {Brassica  oleracea  capitata)  W :  leaves  gnawed 

down  to  midrib  and  veins  and  ends  of  veins  eaten  off. 
2-5  Oct.  '18.  Canterbury  Bell,  Blue  {Campanula  medium)  N^:  leaves  and  petals 

well  devoured. 
6-7  Sept.  '18.    Carrot  {Daucus  Carota):  root  not  attacked  where  covered  by 

skin,  but  cut  end  was  much  gnawed. 
6-11  Sept.  '18.    Celery  {Apium  graveolens)  H^:  leaves  holed  and  their  edges 

gnawed. 
29  Sept.-3  Oct.  '18.   Cherry  {Prunus  [Cerasus]  sp.)  M^:  leaves  not  attacked. 
20-23  Oct.  '18.    Chickweed  {Stellaria  media)  R^:    edges  of    leaves   gnawed 

sUghtly. 
31  Aug.-6  Sept.  '18.    Chrysanthemum,  Garden  {Chrysanthemum  indicum): 

flower  buds  used  as  refuge,  tips  of  petals  apparently  somewhat  nibbled: 

31  Aug.-6  Sept.  '18,  purple  variety:  edges  of  leaves  much  nibbled;  flower 

buds  used  as  refuge,  tips  of  petals  apparently  somewhat  nibbled:  31  Aug.- 

6  Sept.  '18,  white  variety:  leaves  not  attacked;  petals  much  eaten. 
20-24  Sept.  '18.   Clematis,  White  {Clematis  sp.)  K^:  leaves,  a  few  eaten  off  at 

ends  and  edges  gnawed  here  and  there ;  flowers  entirely  devoured. 
23-27  Oct.  '18.    Cluvia  miniata  (Natal):  leaves  not  attacked;  petals  gnawed  a 

little  along  edges. 
15-20  Sept.  '18.    Cornflower  {Centaurea  Cyanus)  .P:  leaves  well  eaten,  only 

midrib  ]eit;  flowers  entirely  devoured. 
29  Sept.-3  Oct.  '18.    Cups  and  Saucers  {Cobaea  scandens)  M^:  petals  nibbled 

a  httle. 

27  Oct.-3  Nov.  '18.    Dandelion  {Taraxacum  oflicinale):  petals  of  ray  florets 

entirely  devoured. 
26-31  Aug.  '18.  Elephant's  Ear,  Pink  {Begonia  sp.):  leaves  much  gnawed  along 

edges  and  also  holed ;  flowers  thoroughly  devoured. 
2-5.  Oct.  '18.  Fern,  Male  {Lastraea  fllis-mas)  0°:  leaves  not  attacked. 
15-20  Sept.  '18.   Feverfew  {Pyrethrum  sp.)  J^:  leaves  gnawed  down  to  midrib; 

flowers  apparently  not  attacked. 
21-28  Sept.  '18.   Fig  {Ficus  Carica):  leaves  not  attacked;  fruit  neglected  when 

whole,  but  cross  section  was  well  gnawed. 
7-15  Oct.  '18.  Fox-glove  {Digitalis  purpurea)  P^:  leaves  holed. 
6-11  Sept.  '18.    Fuchsia,  Crimson  Garden  {Fuchsia  sp.)  H^:  neither  leaves  or 

flowers  were  attacked. 

28  Sept.-2  Oct.  '18.  Geranium,  Scarlet  {Oeranium  sp.)  L^:  petals  eaten  a  httle. 
20-24  Sept.  '18.  Gesnera,  Orange  and  Pink  {Gesnerasp.  )K}:  leaves  not  attacked; 

petals  entirely  devoured. 
24-26  Aug.  '17.    Golden  Rod  {Solidago  sp.)  C^:  leaves  gnawed  at  edges  here 

and  there ;  flowers  apparently  not  attacked. 
2-5  Oct.  '18.   Gooseberry  {Ribes  grossularia)  O^:  leaves  not  attacked. 
11-15  Sept.  '18.   Hawthorn  {Crataegus  oxyeantha)  P:  neither  leaves  ov  flowers 

were  attacked. 
24-31  Aug.  '18.  Hollyhock,  Dark  Crimson  {Althaea  rosea):  leaves  not  attacked; 

flower  buds  used  as  refuge,  petals  apparently  eaten  to  some  extent. 
10-20  Aug.  '18.    Honeysuckle  {Lonicera  sp.)  G^:  leaves  not  attacked;  fruit 

gnawed  considerably. 
7-20  Oct.  '18.   Hydrangea,  Pink  {Hydrangea  sp.)  Q":  neither  leaves  ov  flowers 

were  attacked. 
7-15  Oct.  '18.    Larkspur,  Garden  variety  {Delphinium  sp.)  Q^:  leaves  gnawed 

thoroughly  down  to  midrib. 
3-6  Nov.  '18.   Leek  {Allium  porrum)  T^:  leaves  gnawed  deeply  towards  base. 
6-15  Sept.  '18.    Lettuce,  Cabbage  {Lactuca  sativa):  stem  aljundantly  gnawed 

and  bored;  leaves  of  "heart"  entirely  devoured. 
7-27  Oct.  '18.  Lupin  {Lupinus  polyphyllus)  S^:  leaves  gnawed  to  some  extent. 


of  the  Common  Earwig  (Forficula  auric ularia)  53 

3-17  Nov.  '18.    Mallow  [Malvus  ?  sylvestris):  leaves  holed  and  edges  gnawed 

down  to  veins. 
23  Oct.-17  Nov.  '18.  Marguerite,  White-rayed  [Chrysanthemum  leucanthemum) 

S^,  U^:  petals  of  ray  florets  well  gnawed. 
20-21  Aug.  '17.    Marrow,  Vegetable  (Cucurbita  ovifera)  A^:  leaves  thoroughly 

devoured. 
20-21  Aug.  '17.   Michaelmas  Daisy  (Aster  sp.)  A^,  'i>i^:  leaves  hardly  touched, 

if  at  Sill;  floivers  also  neglected. 
11-15  Sept.  '18.   Mignonette  (Reseda  odorata):  leaves  gnawed  down  to  midrib; 

flowers  attacked  but  slightly  or  not  at  all. 
16-18  Sept.  '18.    Mint  (Mentha  sp.):  leaves,  edges  and  ends  nibbled;  flowers 

entirely  devoured. 
20-23  Oct.  '18.  Xavew  (Brassica  campestris)  R^ :  leaves  holed  and  edges  gnawed 

a  little;  petals  moderately  attacked. 
3-17  Nov.  '18.   Nettle  (Urtica  dioica)  U^:  leaves  well  gnawed  down  to  veins. 
31  Aug. -6  Sept.  '18.   Onion  (Allium  Cepa)  L":  inflorescence  used  as  refuge,  but 

apparently  not  eaten. 
7-15  Oct.  '18.    Pansy  (Viola  tricolor)  P^:  leaves  nibbled  slightly. 
10-20   Aug.    '18.     Parsley,   Garden   (Carum   Petroselinum)   G^:    inflorescence 

nibbled  moderately. 
29   Sept. -3   Oct.   '18.    Peach   (Prunus  [Amygdahis]   sp.)  N^:  leaves  gnawed 

moderately. 
28  Sept.-2  Oct.  '18.   Periwinkle,  Blue  (Vinca  sp.)  L^:  leaves  and  petals  gnawed 

moderatelv. 
22-23  Aug.  '17.    Phlox,  White  (Phlox  Drummondi)  B^:  leaves  apparently  not 

attacked;  petals  much  gnawed  and  pollen  found  in  gut  of  earwigs. 
1-3  Sept.  '17.   Plum  (Prunus  communis)  F^:  fruit  well  eaten. 
23-31  Aug.  '18.    Poppy,  Garden  (Palaver  sp.):  dried  fruits  very  popular  as 

refuges;  some  were  holed  to  obtain  entrance. 
1-18  Sept.  '17.    Potato  (Solanum  tuberosum)  F^:  tuber  in  skin  was  neglected, 

but  slices  were  thoroughly  gnawed. 
28-29  Aug.  '17,  20-23  Oct.  '18.   Primrose,  Evening,  yellow  variety  (Oenothera 

sp.)  D^;  leaves  not  attacked;  petals  eaten  thoroughly;  pods  neglected. 
7-15  Oct.  '18.   Privet  (Ligustrum  vulgare)  Q'^:  leaves  holed  and  edges  gnawed; 

fruits  not  attacked. 
20-21    Aug.   '17.     Radish,    Horse    (Raphanus    sativus)    A^:    leaves    nibbled 

slightly. 
27-29  Aug.  '17.  Raspberry  (Rubus  idaeus)  D":  leaves  not  attacked,  but  earwigs 

assembled  in  crowds  on  their  hairy  undersides. 
22-28  Sept.  '18.    Red  hot  poker  (Kniphofla  sp.)  :  cut  end  of  stem  gnawed; 

lea  ves  and  petals  not  attacked. 
11-15  Sept.  '18.    Rest-harrow  (Ononis  sp.)  P:  apparently  neither  leaves  or 

floivers  were  attacked. 
30-31  Aug.  '17.   Rhubarb  (Rheum  officinale)  W:  leaves  well  gnawed. 
24-26  Aug.  '17.  Rose,  White  garden  variety  (Rosa  sp. )  C^ :  haves  not  attacked ;' 

petals  devoured. 
7-10  Oct.  '18.  St  John's  Wort  (Hypericum  sp.)  P^:  leaves  holed  and  their  edges 

gnawed ;  floioer  buds  not  attacked. 
31  Aug. -6  Sept.  '18.    Scabious,  Crimson  Garden  (Scabiosa  atro-purpurea): 

leaves  much  holed ;  floivers  apparently  not  attacked. 
23-27  Oct.  '18.  Scotch  Kale  (Brassica  oleracea  acephala)  S^:  leaves  holed  a  very 

little ;  curled  margins  a  favourite  refuge. 
10-24  Aug.  '18.    Sea  Kale  (Brassica  oleracea  acephala)  G^:  leaves  holed  and 

gnawed  away  from  edges  to  between  veins. 
6-11  Sept.  '18.   Snapdragon,  Scarlet  (Antirrhinum  sp.):  leaves  gnawed  moder- 
ately; petals  apparently  holed  to  some  extent,  also  used  as  refuge. 


54  Mr  BrincUey,  Further  Notes  on  the  Food  Plants 

23-30  Oct.  '18.    Sow  thistle  [Sonchus  oleaceus):  leaves  holed  slightly;  flower 

huds  not  attacked. 
3-17  Nov.  '18.   Strawberry  (Fragaria  vesca)  W:  leaves  holed  a  little. 
31  Aug.-6  Sept.  '18.   Tomato  (Lycopersicum  esculentum):  leaves  and  vipe  fruit 

gnawed  thoroughly. 
14-15  Sept.  '18.    Valerian,  Red  Garden  ( Valeriana  sp. ) :  edges  of  leaves  gnawed 

moderately ;  petals  entirely  devoured. 
21-24  Aug.  '18.    Vervain,  Blue  (Verbena  sp.):  leaves  nibbled  slightly,  haiiy 

undersides  used  for  assembhng;  petals  entirely  devoured. 
24-31  Aug.  '18.    Vetch,  Mauve  and  White  garden  varieties  (Vicia  sp.):  leaves 

attacked  very  slightly,  if  at  all;  petals  entirely  devoured. 
23  Oct.-3  Nov.  '18.    Violet,  Single  and  Double  garden  varieties  {Viola  sp.): 

leaves  holed  and  edges  gnawed  moderately. 
3-17  Nov.  '19.    Wartweed  (Euphorbia  helioscopia)  T^:  edges  of  leaves  gnawed 

very  shghtly. 
15-18  Sept.  '18.   Wormwood  (Artemisia  sp  ):  leaves  not  attacked. 


These  observations  are  of  course  subject  to  tbe  drawback  that 
in  captivity  animals  which  normally  feed  daily  may  take  unusual 
food  with  apparent  eagerness  because  no  other  is  available;  but 
the  above  record  probably  indicates  normal  preferences  over  a 
certain  range  of  common  plants,  and  also  that  some  are  disliked 
by  earwigs;  thus  Wartweed  was  left  entirely  untouched  for  many 
days  in  the  absence  of  any  other  food,  the  animals  attacking  potato 
tuber  ravenously  as  soon  as  this  was  substituted.  It  seems  natural 
that  such  stiff  and  dry  foliage  leaves  as  those  of  Raspberry,  Haw- 
thorn, and  Cherry,  should  escape  attack,  and  there  is  no  doubt  that 
the  more  succulent  leaves  are  preferred.  The  list  of  plants  affords 
some  information  which  may  facilitate  the  destruction  of  earwigs 
when  they  become  a  pest  by  the  indications  obtained  as  to  plants 
which  are  popular  as  refuges,  and  also  by  the  mode  in  which  the 
attack  on  leaves  is  made;  thus,  some  leaves  seem  to  be  attacked 
by  holing  as  well  as  by  gnawing  along  the  edges,  and  others  only 
by  the  latter  method.  There  is  no  doubt  that  earwigs  have  pre- 
ferences among  the  common  plants  of  a  flower  or  vegetable  garden, 
and  that  if  numerous  they  are  likely  to  become  a  pest.  In  certain 
cases,  as  for  instance,  chrysanthemums,  the  actual  damage  done 
seems  to  be  exaggerated  by  common  report. 

Since  the  epitome  of  recent  literature  on  the  subject  in  my 
previous  paper  {Proc.  Camb.  Phil.  Sac,  xix,  Part  4,  1918,  p.  170) 
was  written.  The  Review  of  Applied  Entomology  has  recorded 
attacks  on  beets  and  sugar-beets  in  Denmark  sufficiently  serious 
to  obtain  mention  by  Lind  and  others  in  their  Report  on  Agri- 
cultural Pests  in  1915  {Beretning  fra  Statens  Forsogsvirksomhed  i 
Plantekultur,  Copenhagen,  1916,  pp.  397-423). 

As  regards  the  carnivorous  habit  of  F.  auricularia,  lean  roast 
mutton  without  other  food  was  given  for  several  days  to  the  ear- 
wigs under  observation  in  1918  and  was  gnawed  sparingly,  while 


of  the  Common  Earwig  (Forficula  auricularia)  55 

mutton  suet  substituted  for  it  was  eaten  readily  and  extensively. 
In  the  Journal  of  the  Bombay  Natural  History  Society,  xxvi,  No.  2, 
May,  1919,  p.  688,  F.  P.  Connor  records  an  unnamed  earwig  at 
Amara  catching  moths  in  its  forcipes  and  in  one  case  nibbUng  its 
prey.  F.  Maxwell  Lefroy  {Indian  Insect  Life,  p.  52)  remarks: 
"The  function  of  the  forcipes  is  a  mystery  that  will  be  cleared  up 
only  when  their  food  habits  and  general  hfe  are  better  under- 
stood." They  are  very  possibly  "frightening"  as  well  as  defensive 
organs.  Pemberton  {Hawaiian  Planters'  Record,  Honolulu,  xxi, 
No.  4,  Oct.  1919,  pp.  194-221)  mentions  the  benefit  to  cane  fields 
arising  from  the  destruction  of  the  leaf-hopper  parasite  Perkin- 
siella  optabilis  by  the  black  earwig  Chelisoches  morio. 

The  importance  of  nocturnal  observations  on  the  feeding  habits 
of  Forficula  auricularia  to  a  satisfactory  understanding  of  the 
economic  effects  of  this  insect  in  gardens,  urged  in  my  previous 
paper,  may  be  referred  to  again. 


56     Mr  Darwin,  Lagrangian  Methods  for  High  Speed  Motion 

Lagrangian  Methods  for  High  Speed  Motion.  By  C.  G.  Darwin. 
[Read  8  March  1920.] 

1.  In  the  later  developments  of  Bohr's*  spectrum  theory,  it 
is  necessary  to  calculate  the  orbits  of  electrons  moving  \vath  such 
high  velocities  that  there  is  a  sensible  increase  of  mass.  The  selection 
of  the  orbits  permitted  by  the  quantum  theory  almost  necessitates 
the  treatment  of  such  problems  by  Hamiltonian  methods.  Working 
on  these  lines  Sommerfeldf  and  others  have  calculated  with  a  very 
high  degree  of  success  those  spectra  which  involve  the  motion  of 
a  single  electron.  But  the  application  of  the  Hamiltonian  function 
involves  a  knowledge  of  the  momentum  corresponding  to  any 
generalized  coordinate,  and  in  the  formulation  of  most  problems 
the  momenta  are  not  known  a  priori  but  must  be  calculated  from 
the  corresponding  velocities.  In  other  words  the  formation  of  the 
Hamiltonian  function  must  in  general  be  preceded  by  that  of  the 
Lagrangian.  An  exception  occurs  in  precisely  the  problems  referred 
to  above;  for,  the  electromagnetic  theory  furnishes  directly  values 
for  the  momentum  and  kinetic  energy  of  a  moving  electron  in 
terms  of  its  velocity,  and  the  velocity  can  be  eliminated  between 
them  so  as  to  obtain  the  Hamiltonian  function.  But  in  even  slightly 
more  complicated  cases  this  simple  relation  is  destroyed — thus  the 
problem  of  a  single  electron  in  a  constant  magnetic  field  can  only 
be  solved  by  introducing  the  artificial  conception  of  rotating  axes 
• — and  in  general  it  will  be  necessary  to  follow  the  direct  course  of 
finding  the  Lagrangian  function  in  terms  of  the  generalized  velocities, 
and  then  deducing  from  it  the  momenta  and  the  Hamiltonian 
function  in  the  usual  way. 

If  more  than  one  particle  is  in  motion  another  difficulty  enters. 
For  the  interaction  of  two  moving  particles  depends  on  a  set  of 
retarded  potentials  and  the  effect  of  the  retardation  is  readily  seen 
to  be  of  the  same  order  as  the  increase  of  mass  with  velocity.  The 
calculation  of  the  retardation  can  only  be  carried  out  by  expansion 
and  so  the  results  are  only  approximate.  This  is  not  surprising  since 
the  methods  of  conservative  dynamics  cannot  apply  to  such  effects 
as  the  dissipation  of  energy  by  radiation,  effects  inevitably  required 
•  by  the  electromagnetic  theory,  though  they  do  not  occur  in  actuality. 
We  can  also  see  from  the  fact  that  these  radiation  terms  are  of 
the  order  of  the  inverse  cube  of  the  velocity  of  light,  that  it  will 
be  useless  to  expand  beyond  the  inverse  square. 

*  N.  Bolir,  Kgl.  Dan.  Wet.  SelsL,  1918. 

t  A.  Sommerfeld,  Ann.  Phys.,  vol.  51,  p.  1,  1916. 


Mr  Darwin,  Lagrangian  Methods  for  High  Speed  Motion     57 

2.  We  first  consider  the  motion  of  a  single  electron  in  an 
arbitrary  electric  and  magnetic  field  varying  in  any  manner  with 
the  time  and  position.  If  m  is  the  mass  for  low  velocities,  the 
momentum  is  known  to  be  mv/^,  where  ^  =  V  1  —  v^jc^.  Starting 
from  this  we  have  quasi-Newtonian  equations  of  motion  of  the 
type 

lir*}-^^  '^■^'- 

The  force  F^.  is  given  from  the  field  E,  H  as  the  vector  eE  +  ^  [v,  H], 

where  v  is  the  velocity  vector  of  the  particle's  motion.    E  and  H 

can  be  expressed  in  terms  of  the  scalar  and  vector  potentials  in 

1  3A 
the  form  E  =  —  grad  (t>  ~  ^.^^  and  H  =  curl  A. 

C  ct 

Then  if  r^  is  the  vector  x,  y,  z  we  have  as  the  vector  equation  of 
motion 

It  \t  '4  ^~'' ^''^ '^  ~  c  W  +  c  ^'1'  '""'^ ^^  •••(^■^^' 

where  ^^  =  V  1  -  V/C^. 

Let  q  be  any  one  of  three  generalized  coordinates  representing 
the  position  of  the  particle.    Take  the  scalar  product  of  (2-2)  by 

i=r^.    Then  since  ^  =  -^,  we  have 
oq  cq       cq 


dii    d   (nil  .  )\       d   (m 


dq'dtX^^'^^U      dt\^. 


dt  dq      dq 

'9r    _    .    ,\_  d4> 


where  "Wq  =  j;^  —  ^  ^^^  Lagrangian  operator. 


Again  -  gj  [J-,  grad  .^ j  =  -  e^  ^  =  e^Bc/,. 

The  remainder  can  be  reduced  to 

CV''~dq)~c[dq'lt)  ^-'^^' 

dA     dA     dA  .      dA  .      8A . 

where  -j7  =  ^7  +  ^^+^-y+'^^ 

dt       dt       ox  Cy  '^       oz 


58     Mr  Darwin,  Lagrangian  Methods  for  High  Speed  Motion 

and  so  is  the  total  change  of  A  at  the  moving  particle.    (2-3)  can 

be  reduced  to  —  ^  IBq  (fi,  A), 

Thus  the  whole  equation  of  motion  can  be  derived  from  a 
Lagrangian  function 

L=-  m,C^^,  -  e[<j>  +  g,  (ii,  A)  (24). 

This  is  valid  for  any  fields  of  force  including  explicit  dependence 
of  ^  and  A  on  the  time.  The  first  term  in  L,  which  reduces  to  the 
kinetic  energy  for  low  velocities,  differs  from  it  in  general.  It  is 
very  closely  connected  with  the  "world  line"  of  the  particle. 

3.  To  treat  of  the  case  where  several  moving  particles  interact 
we  shall  start  by  supposing  that  there  is  a  second  particle  present 
undergoing  a  constrained  motion  so  that  its  coordinates  are  imagined 
to  be  known  functions  of  the  time.  The  same  will  then  be  true  of 
the  potentials  it  generates.  The  motion  of  e^  will  then  be  governed 
by  (2-4)  if  ^  and  A  are  expressed  in  terms  of  the  motion  of  e^.  These 
potentials  are  given  by 

/   _  ^2 a   _  ^ ^2  /o.-i  \ 

In  these  expressions  r^  =  (ig  —  ij)^  and  the  values  are  to  be  retarded 
values.  If  the  time  of  retardation  be  calculated  and  the  result 
substituted  in  (3-1)  we  obtain 

/_e2,     62    \i^^+{i^,r^-Ti)      (f^,  r^  -  i,)^\     ._e.,i^ 

where  now  ij,  la  refer  to  the  same  instant  of  time,  cf)  is  an  approxi- 
mation valid  to  C"^,  but  the  value  of  A  has  only  been  found  to 
the  degree  C~^  on  account  of  the  further  factor  C~^  in  (2-4)  which 
is  to  multiply  it.  Then  substituting  in  (2-4)  we  obtain 

r  _      ,^  P2/P       ^1^2     6162  (r2^+  (r2,r2-ri)  -  2  (fi,  f^) 
L--  m,C  Id,  --y-^,  I 


The  equations  of  motion  are  unaffected  by  adding  to  L  the  expres- 
sion -  mgC^^a  +  ^  ^2  ^'^"^'~^'^-   The  first  is  a  pure  function  of 

the  time  and  so  contributes  no  terms  to  the  equations  of  motion. 
The  second  contributes  nothing  because  for  any  function  /  we  have 


Mr  Danvin,  Lagrangian  Methods  for  High  Speed  Motion     59 
The  new  form  of  L  then  reduces  to 

L  =  -  m,C-^^,  -  m,C^^,  -  f  +  |gi  j^^^ 

I  (ri,ra-ri)(fa,r2-ri)|    ^^.^^^ 


From  the  complete  symmetry  of  this  form  the  roles  of  e^  and  eg  ^^^-y 
be  interchanged.  Further  from  the  covariance  of  the  operator  IB 
for  point  transformations,  both  may  be  included  in  the  dynamical 
system,  so  that  if  q  is  any  generalized  coordinate  involving  both 
Tj  and  ig,  the  equations  of  motion  will  be  of  the  form  "313 gL  =  0. 

For  the  sake  of  consistency,  as  the  last  term  in  (3-4)  is  only  an 
approximation  valid  to  C~^,  the  first  two  should  be  expanded  only 
to  this  power.  The  first  term  will  give 

-  m^C^  +  lm,i^^  +  g^  mj^\ 

Generalizing  our  result  to  the  case  of  any  number  of  particles 
in  any  external  field  we  have 

L  =  ^lm,i,^  +  2  g^,  m,i,^  -  He.cf.  +  2  ^,  (r,A)  -  SS  '^^ 

+  si;  ^^  I^AiA^  +  (ri,r2-ri)  (^2>r2-ri))  ^   /3.5)_ 

The  double  summations  are  taken  counting  each  pair  once  only. 
4.    The  transition  to  the  Hamiltonian  now  follows  the  ordinary 

r)  T 

rules.   We  find  momenta  f  =  -^  and  solve  for  the  g-'s  in  terms  of 

the  2^'s.  This  can  be  done  in  spite  of  the  cubic  form  of  the  equations 
in  the  g's  by  use  of  the  approximation  in  powers  of  C.  The  Hamil- 
tonian function  will  then  he  H  =  Hpq  —  L  and  the  equations  of 

motion  will  be  the  canonical  equations  q  =  ^—,  p  =  —  -o— •    K  Pi 

be  the  momentum  corresponding  to  Tj ,  the  Hamiltonian  in  these 
coordinates  will  be 

«- ^  2lJ  -  ^^  sit' +  ^^'^  -  ^  CS^  <""*»  + ^^  t 


_  ss      ^1^2       [(Pi>P2)  ^  (Pi,r2-ri)  (P2,r.,-ri) 

All  the  applications  of  general  dynamics,  such  as  the  Hamilton 
Jacobi  partial  differential  equation,  follow  from  this.  As  in  ord